Levy-Khintchin Theorem for best simultaneous Diophantine approximations
Yitwah Cheung Nicolas Chevallier



Lévy-Khintchin Theorem for best simultaneous Diophantine approximations
    Yitwah Cheung, Nicolas Chevallier
    
=======================================================================





We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. 
The first result is the Lévy-Khintchin theorem about the almost sure growth rate of the denominators of the convergents. The second result is a theorem of Doeblin and of Bosma, Jager and Wiedijk about the almost sure limit distribution of the sequence of products  q_n(q_nθ,) where the q_n's are the denominators of the convergents associated with the real number  θ by the ordinary continued fraction algorithm. Besides these two main results, we show that when d≥ 2, for almost all vectors θ∈^d, lim inf_n→∞ q_n+k(q_nθ,^d)^d=0 for all positive integers k, where (q_n)_n∈ is the sequence of best approximation denominators of θ. 







Nous étendons deux résultats sur le développement en fraction continue ordinaire aux meilleures approximations diophantiennes simultanées  de vecteurs ou de matrices.
Le premier résultat est le théorème de Lévy-Kintchin sur le taux de croissance presque sûr des dénominateurs des réduites. Le second est un théorème de Doeblin et de Bosma, Jager et Wiedijk sur la distribution presque sûre de la suite des produits q_n  (q_n θ, ) où les q_n  sont les dénominateurs des réduites associées au nombre réel θ par l'algorithme des fractions continues ordinaires. En dehors de ces deux  résultats principaux, nous montrons que lorsque d ≥ 2, pour presque tous les vecteurs θ∈ ^ d, lim inf_ n →∞ q_n+k (q_n θ, ^d)^d = 0 pour tous les entiers positifs k, où (q_n) _ n ∈ est la suite des dénominateurs des meilleures approximations de θ.






approximations diophantienne, theorie ergodique, flot diagonal

11J13,11J83,11K55,37A17



§ INTRODUCTION


In 1936, Aleksandr Khintchin showed that there exists a constant γ such that the denominators (q_n)_n≥ 0
of the convergents of the continued fraction expansions of almost all real
numbers θ satisfy

    lim_n→∞q_n^1/n=γ

(see <cit.>). Soon afterward, in
<cit.>, in the footnote page 289, Paul Lévy gave the explicit value
of the constant,

    γ=expπ^2/12ln 2.


In <cit.>, published in 1940, among many other results about the ordinary continued fraction expansion, Wolfgang Doeblin stated the following result: 
 for almost all real numbers θ,

    lim_n→∞1/n*card{0≤ k<n:q_k(q_kθ,ℤ)≤ t}=g(t)

for all t∈[0,1], where 

    g(t)=∫_0^t1/2ln 21-|
    1-2s|/s ds.
 
Doeblin only sketched the proof of this result and it is difficult to reconstitute a complete proof from his paper. In <cit.>, Iosifescu gave a very interesting account of Doeblin's paper.  Doeblin's outstanding article was not immediately noticed, so the above result was first called the Lenstra conjecture, now it is often called the Doeblin-Lenstra conjecture. A complete proof of the Doeblin-Lenstra conjecture was given in 1983 by Wieb Bosma, Hendrik Jager and Freek Wiedijk, see  <cit.>. Later, Jager proved variants of this result, in
particular with the quantity q_k+1(q_kθ,ℤ) instead of
q_k(q_kθ,ℤ) (see <cit.> for more results).

The aim of our work is to extend to the best simultaneous Diophantine approximations, both the Lévy-Khintchin result and the Jager version of the Doeblin, Bosma, Jager and Wiedijk result.
We choose the Jager version because one of the striking difference between one-dimensional and multidimensional Diophantine approximations is that the classic one-dimensional lower bound q_k+1(q_kθ,ℤ)≥ 1/2  no longer holds in the multidimensional setting; this explains for example that the only singular vectors are the rational numbers in dimension one.   


Let d and c be two positive integers. Suppose ℝ^d and
ℝ^c are endowed with the standard Euclidean norms ‖
.‖ _ℝ^d and ‖ .‖ _ℝ^c. We prove


 There exists a constant L_d,c such that for almost all
matrices θ∈_d,c(ℝ),

    lim_n→∞1/nln‖ Q_n(θ)‖
    _ℝ^c     =L_d,c ,
    lim_n→∞1/nln((θ Q_n(θ),ℤ
    ^d))       =-c/dL_d,c

where Q_n(θ)∈ℤ^c, n≥ 0, is the sequence of best
Diophantine approximation denominators of θ associated with the norms
‖ .‖ _ℝ^d and ‖ .‖
_ℝ^c.


(See Section 2.1 for the definition of best Diophantine approximation denominators).

 Theorem <ref> is stated for standard Euclidean norms but it holds for all pairs of Euclidean norms on ^d and ^c with the same constant L_d,c (see Theorem <ref>). We are convinced that Theorem <ref> is valid for general norms but we do not know whether the constants L_d,c depend on the norms  (see Section <ref>).


For a matrix θ in _d,c(ℝ), let us denote β_n
(θ)=‖ Q_n+1(θ)‖ ^c _ℝ^c(θ
Q_n(θ),ℤ^d)^d and for a real number a, let us denote  δ_a  the Dirac measure at a.


1. There exists a probability measure ν_d,c on
ℝ such that, for almost all matrices θ∈_d,c(ℝ
), ν_d,c is the weak limit of the sequence of probability measures

    1/n∑_k=0^n-1δ_β_k(θ).

2. The support of the
measure ν_d,c is included in a  bounded interval, and contains 0 provided
that c+d≥ 3.


The Lévy-Khintchin result has already been extended to multidimensional
settings. For instance, for almost all θ in ℝ^d, the
denominators (J_n(θ))_n≥ 0 of the Jacobi-Perron expansion of
θ satisfy lim_n→∞1/nln J_n(θ
)=c_d for some constant c_d (see <cit.>). The common proofs of such results use ergodic theory. The one-dimensional
Lévy-Khintchin result can be proven with the Birkhoff ergodic theorem, while
the growth rate of the Jacobi-Perron denominators can be derived from the Oseledec
multiplicative ergodic theorem. In both cases, the proof depends on the
existence of an underlying dynamical system: the Gauss map or the
Jacobi-Perron map (see <cit.> for many examples of these kinds of
maps). However, no such map associated with best Diophantine approximations is
known when d+c≥ 3.  One  way to circumvent this problem is
to use the left action of the diagonal flow

    g_t=
    [  e^ctI_d        0;        0 e^-dtI_c ]∈ (d+c,ℝ), t∈,

on the space of unimodular lattices ℒ_d+c= (d+c,ℝ
)/ (d+c,ℤ).  This idea can be traced back to Hermite <cit.> and has been used extensively in Diophantine approximation since the work of Dani <cit.>; see for instance <cit.>.
Among many works using  diagonal flows and  homogeneous dynamics in Diophantine approximations, we indicate three works that illustrate the variety of the subjects tackled, the survey <cit.> by Ghosh,   <cit.> by An, Guan and Kleinbock on weighted badly approximable numbers and <cit.> by Haynes and Marklof on extensions of the three distances theorem. 
In the same vein, the parametric geometry of numbers introduced recently by Schmidt and Summerer <cit.> and complemented by Roy <cit.>, studies the evolution of the successive minima of a lattice under the action of the diagonal flow. This led to many  results on Diophantine exponents; see for instance <cit.>.


In <cit.>, the diagonal flow is used
to prove that the sequence of best Diophantine approximation denominators of
almost all θ in _d,c(ℝ) satisfies

    limsup_n→∞1/nln‖ Q_n(θ)‖
    _ℝ^c≤ K_d,c

for some constant K_d,c. When c=1, it is also possible to derive this inequality
from a theorem of W. M. Schmidt (see
<cit.>).

As in the aforementioned works, the diagonal flow (g_t) is the main tool. However, unlike e.g. <cit.>, we introduce an important new ingredient: a co-dimension one submanifold transverse to the flow. It should be noticed that many works on continued fraction algorithms also use transversals, we mention two such works.
In this work,  the flow (g_t) is the main tool.
Together with the flow, an important ingredient of our work is a  co-dimension 1 submanifold S transverse to the flow and the first return map
associated with the flow. 
Such
transversals have been widely used and we  mention three closely related works.


Firstly, P. Arnoux and A. Nogueira in <cit.>, have used
transversals to naturally obtain   invariant measures associated with
multidimensional continued fraction algorithms. Furthermore, in the case of
the one dimensional continued fraction algorithm, their approach led to an
interpretation of  Lévy's constant as the average return time of the
flow on the transversal.



Firstly, the transformation induced on a well-chosen sub-interval by an interval
exchange transformation T is an interval exchange transformation T̂
defined with the same number of intervals. The Rauzy-Veech continued fraction algorithm is the map  R:T ↦T̂ where T is an interval exchange transformation defined on an interval of length 1 and T̂ is renormalized so that it is defined on an interval of length 1 (see <cit.>).   In <cit.>, W. Veech  proved
that R admits a unique absolutely continuous invariant measure up
to a scalar multiple, using a bimeasureably invertible extension R̂ of the map R.  In turn, this extension is constructed as the first return map of a “diagonal flow” on a transversal. It should be noticed that at the same time, H. Masur
proposed a very similar construction in <cit.>. 
Thirdly,
 A. Haas, extended both the result of Lévy-Khintchin and  the result of Doeblin, Bosma, Jager and Wiedijk to  finite aera non compact hyperbolic surfaces, see <cit.>. His extensions are expressed in terms of depth of the geodesic excursions into the cusp. Shifting from the surface to an appropriately normalized Fuchsian group
representing the surface, A. Haas found a direct analogue to both results where the rational numbers are replaced by a subset of the real numbers associated with the Fuchsian group. The proofs use the ergodicity of the geodesic flow and a transversal. 


Secondly, A. Haas proved analogs of the result of Doeblin, Bosma, Jager and Wiedijk and of the Lévy-Khintchin result in the setting of excursions of geodesic into the neighborhood of a cusp on a finite area hyperbolic 2-orbifold, see <cit.>. A. Haas used the ergodicity of the geodesic flow and a transversal in the unit tangent bundle that projects on a finite union of geodesic segments of the hyperbolic plane.
 
In our case,  up to a negligible set of flow trajectories, the transversal is the set of unimodular lattices  whose  first two  minima  are equal (see
Section 3.1). The first return map of the flow in the transversal  play the role of an invertible extension of the missing Gauss map. It is crucial to observe
that the visiting times of the transversal by the flow are given by a simple
formula involving best simultaneous Diophantine approximations; see
Lemma <ref>.  Making use of the Birkhoff ergodic theorem, this observation leads to a
Lévy-Khintchin result in the space of lattices and to a formula close to
the Arnoux-Nogueira interpretation of the Lévy's constant (see <cit.>):

    L_d,c=d/μ_S(S)∫_Sτ dμ_S=d×μ
    (ℒ_d+c)/μ_S(S)

where μ is the invariant measure in the space of lattices, μ_S the
invariant measure induced by the flow on the transversal S and τ the
return time to S, see Theorem <ref> and Corollary <ref>.

The second step of the proof of Theorem <ref> consists in converting an
almost sure result in the space of lattices ℒ_d+c into an almost sure result in
_d,c(ℝ). To achieve this goal, we prove a general result,
Theorem <ref>, which might be of independent interest. At
first sight, this result might appear as an easy consequence of the following standard fact: the
set of lattices associated with the matrices θ in _d,c
(ℝ), is the expanding direction of the flow g_t. However, an example
shows that Theorem <ref> depends on some properties of the transversal; see
Section <ref>.

When d=1 or 2 and c=1, the submanifold S and  the measure μ_S can be entirely calculated 
(see Section <ref>). When d=c=1, thanks to  Siegel formula
giving the volume of the modular space (2,)/(2,),
computing  Lévy's constant L_1,1=lnγ is easy; it is even
possible to determine the first return map to the transversal. It
turns out that this first return map is a 2-fold extension of the
natural extension of the Gauss map (see Subsection <ref>).
However, when d=2 and c=1, the calculation of
L_2,1 leads to a seven-tuple integral and we have only succeeded in reducing it
to a triple integral which  can be evaluated numerically (see Subsection <ref>).

When d=c=1, the double inequality 1/2≤ q_n+1(q_nθ,ℤ)≤ 1 shows that the behaviors of the two sequences
(1/nln q_n)_n and (-1/nln(q_nθ,ℤ))_n are the
same and each of the limits in Theorem <ref> implies the other.
When d is greater than or equal to two, no such double inequality exists. Indeed, it
has been proven in <cit.> that when c=1 and d≥ 2,

    liminf_n→∞q_n+1(q_nθ,ℤ^d)^d=0

for almost all θ in ℝ^d. Observe that  Theorem <ref> implies this latter result; it is an immediate consequence of the
fact that 0 is in the support of the measure ν_d,c. Hopefully for the proof of
Theorem <ref>, the weaker inequality

    (θ Q_n(θ),ℤ^d)≥1/‖ Q_n
    (θ)‖ _ℝ^c^c/dln‖ Q_n(θ
    )‖

which holds almost surely by the convergence part of the Khintchin-Groshev
theorem (see <cit.>), is enough to link both of the limits in Theorem <ref>.

In  Section 9, we extend the aforementioned result of <cit.>
to best simultaneous approximations of matrices. Our proof leads to the
stronger result

    liminf_n→∞‖ Q_n+k(θ)‖
    _ℝ^c^c(θ Q_n(θ),ℤ^d)^d=0

for almost all θ in _d,c(ℝ) and all k∈ℕ.

Obviously, if the matrix θ is badly approximable, then

    liminf_n→∞‖ Q_n+k(θ)‖
    _ℝ^c^c(θ Q_n(θ),ℤ^d)^d>0

because by definition, liminf_n→∞‖ Q_n(θ)‖
_ℝ^c^c(θ Q_n(θ
),ℤ^d)^d>0. When d=2 and c=1, we prove that the set of
θ such that, liminf_n→∞q_n+1(q_nθ
,ℤ)^2>0, is not reduced to the set of badly approximable vectors; see Proposition <ref>.

Overview of the paper. The notations are collected in Section 2. 
The third section is devoted to best Diophantine approximations, to minimal vectors and  to the relations between these two notions. 
The fourth section is devoted to the definition of the transversal S and to another transversal S'. The key point of this section is the expression of visiting times of the transversals by the flow trajectories  with the minimal vectors and the best approximations (Lemmas <ref> and <ref>). The induced measure on the transversal is defined in Section 5. In Section 6, Theorem <ref>  is an important result of the paper, it is a version of Theorem <ref>  in the space of lattices. 
In Section 7, when c=1, a parametrization of the transversal S is described  together with a formula for the induced measure. Furthermore, the two cases d=1 and 2 are considered. Section 7 is not necessary for the proof of Theorems <ref> and <ref>.
In Section 8, we give the general result, Theorem <ref>, which allows to convert the almost sure convergence in the space of  lattices proved in Theorem <ref>, to an almost sure convergence in the space of matrices M_d,c() needed in Theorems <ref> and <ref>. 
In Section 9, we study the behavior of the quantities q_n+k^c(θ)r_n^d(θ) and finish the proof of Theorem <ref>. At last, we ask a few miscellaneous questions in Section 10.



Historical Note: Lévy's proof does not rely on the ergodic theorem, which
was not known for non-invertible maps at that time. A proof of the Birkhoff ergodic
theorem for non-invertible maps was given by Frédéric Riesz in 1945
(see <cit.>) and then a proof of Lévy's theorem using the ergodic theorem
was given in <cit.>. The authors would like to thank Vitaly
Bergelson for bringing <cit.> to their attention.



§ NOTATION

Let d and c be two positive integers.


 §.§ Vectors and distances

For a positive integer n, let ‖·‖ _ℝ^n denote the usual
Euclidean norm on ℝ^n and let B_ℝ^n(u,r) denote the ball of center u and radius r associated with the norm ._ℝ^n.

Let us denote ‖ .‖ _d,c the norm defined on ℝ^d+c by
 ‖ (u,h)‖ _d,c=max{‖
u‖ _ℝ^d,‖ h‖ _ℝ^c} and let B_d,c(X,r) denote the ball of center X and radius r associated with the norm ‖ .‖_d,c.




For X=(u,h) in ℝ^d+c, let  |  X|  _-=
h  _ℝ^c  denote the height of the vector X and
let |  X|  _+=  u  _ℝ^d denote the norm of the
projection of X in the horizontal space. We also denote X_+=u and
X_-=h the   horizontal and vertical components of X.


For a vector X in ℝ^d+c, let  C(X) denote  the closed cylinder in ℝ^d+c

    C(X)=B_ℝ^d(0,| X| _+
    )× B_ℝ^c(0,| X| _-),

and if Y is another vector, let C(X,Y) denote  the closed cylinder

    C(X,Y)=B_ℝ^d(0,| X| _+
    )× B_ℝ^c(0,| Y| _-).

 Observe that in the basic case d=2 and c=1, C(X) is a usual cylinder with height 2 × |X|_- and radius |X|_+.

    < g r a p h i c s >
 

In all cases, we denote by (x,y) the distance associated with the underlying norm between the two points x and y and by (x,A) the distance between the point x and the set A.



 §.§ Matrices




 We fix a norm on _d+c(ℝ).  All the distances and balls in
the  space of matrices are associated with this norm.   This norm will be used as a tool in the proofs and when the choice of this norm is important we will specify it. When E is a subset of
_d+c(ℝ), let B_E(x,r) denote the set of matrices y in E such that (x,y)≤ r.

Let I_n denote the identity matrix in _n(ℝ).
Let 

    g_t=
    [  e^ctI_d        0;        0 e^-dtI_c ]∈ (d+c,ℝ),

t∈ℝ, denote the standard diagonal flow,
E_-= {0}×ℝ^c denote the contracting direction of the flow and
E_+=ℝ^d ×{0} denote the expanding direction of the flow. We also refer to E_+ as the horizontal subspace and to E_- as the vertical subspace.



Let  ℋ_>0 denote the subgroup of all matrices M_θ=(
[ I_d  -θ;   0 I_c ]), θ∈_d,c(ℝ),  and let 𝕋_d,c denote its image in the space of unimodular lattices
ℒ_d+c.
In the same manner, let ℋ_< denote the subgroup of (d+c,ℝ) of matrices of
the form

    (
    [ I_d   0;   B I_c ])

and let ℋ_≤ denote the subgroup of (d+c,ℝ) of
matrices of the form

    (
    [ A 0; B C ])

where A∈(d,ℝ), B∈ M_c,d(ℝ) and C∈(c,ℝ).

Observe that for any M∈ℋ_<, g_tMg_-t→ I_d+c when t tends to +∞ and that the distance from g_tMg_-t to I_d+c is bounded when t∈ℝ_+ for all M∈ℋ_≤.


 §.§ Lattices

Let  ℒ=ℒ_d+c denote the space
of (d+c)-dimensional unimodular lattices in ℝ^d+c. We identify ℒ_d+c with 
(d+c,ℝ)/ (d+c,ℤ).



For θ∈_d,c(ℝ), let Λ_θ be the lattice in ℝ^d+c generated by the columns of the matrix
M_θ, i.e.,  Λ_θ=M_θℤ^d+c. 


Suppose ℝ^n is equipped with a norm ‖.‖. For a lattice Λ and an integer i∈{1,...,n}, let λ_i(Λ,.) denote the i-th minimum of the lattice Λ with respect to the norm ‖.‖, i.e.,

    λ_i(Λ,‖.‖)=inf{λ>0:B(0,λ)∩Λ contains at least i independent vectors}.

Observe that λ_1(Λ) is the length of the shortest nonzero vector in Λ. When there is no ambiguity about the norm, we write λ_i(Λ) instead of λ_i(Λ,‖.‖).

We fix a measure μ  on ℒ_d+c  invariant by the (d+c,ℝ) left action. Recall that μ is unique up to a multiplicative constant.







§ BEST APPROXIMATIONS




 §.§ Best Diophantine approximations


Best approximation vectors have been introduced since a long time inside
proofs in an non-explicit form. 
To our knowledge, C. A. Rogers in 1951 <cit.> was the first to define best
Diophantine approximations. In fact, he defined the best Diophantine approximations associated with the sup norm.

Lagarias was the first to have undertaken a systematic study of the best Diophantine approximations, see  <cit.>. For instance, Lagarias proved that it is not possible to conciliate the unimodularity and the best approximation property (see <cit.> and <cit.>). This explains why a multidimensional continued fraction algorithm never satisfies the best approximation property when this algorithm is defined by a piecewise unimodular homographic generalized Gauss map.   

The connection between best simultaneous Diophantine approximations and the diagonal flow in the space of 
lattices goes back at least to Hermite, see <cit.>. In <cit.>, based on  Hermite's idea, Lagarias defined the Minkowski multidimensional continued fraction expansion  which gives all the “Hermite best approximations” to a given θ∈^d (see also <cit.> for a deep study of the case d=c=1).

For more details and the history of best Diophantine approximations see the surveys  <cit.>. 



Let θ∈_d,c(ℝ). 1. A nonzero vector
Q∈ℤ^c is a best simultaneous Diophantine approximation
denominator of θ (with respect to the norms ‖.‖_ℝ^d and ‖.‖_ℝ^c), if for all nonzero U in ℤ^c,

    ‖ U‖ _ℝ^c     <‖ Q‖
    _ℝ^c⇒(θ Q,ℤ^d)<(θ U,ℤ^d)  and
    ‖ U‖ _ℝ^c     =‖ Q‖
    _ℝ^c⇒(θ Q,ℤ^d)≤(θ U,ℤ^d).

2. An element (P,Q) in ℤ^d×ℤ^c is a
best Diophantine approximation vector of θ if Q is a best
simultaneous Diophantine approximation denominator of θ and if

    ‖θ Q-P‖ _ℝ^d=(θ Q,ℤ
    ^d).




If Q=0 is the only solution in ℤ^d of the equation θ Q=0 ^d, the set of best Diophantine approximation denominators of
θ is infinite. Numbering the set of best approximation denominators in
ascending order of the norm q=‖ Q‖ _c, we obtain two
sequences

    q_0=q_0(θ)=‖
    Q_0(θ)‖ _c=λ_1(ℤ^c)<q_1=q_1(θ)=‖
    Q_1(θ)‖ _c<...<q_n=q_n(θ)=‖ Q_n
    (θ)‖ _c<....

and 

    r_0=r_0(θ)=(θ Q_0,ℤ^d)>r_1=r_1(θ)=(θ Q_1,ℤ^d)>...>r_n=r_n(θ)=(θ Q_n,ℤ^d)>....


When d=c=1, by the best approximation property of the ordinary continued fraction expansion (see <cit.> Theorem 182), the integers q_0
,q_1,...,q_n,... are the denominators of the convergents of the ordinary continued fraction
expansion of θ. The only slight difference is that in the ordinary
continued fraction expansion, it can happen that q_0=q_1=1.   Unless d=c=1, the sequence (V_n=(P_n,Q_n))_n of best approximation vectors does not satisfy a linear recurrence relation of the form V_n=a_n,1V_n-1+…+a_n,d+cV_n-d-c, but   satisfies weak forms of well-known properties of the one-dimensional continued fraction expansion, see Lemmas <ref> and <ref> below.



 §.§ Minimal vectors in lattices


The notion of minimal vector goes
back to Voronoi. He used minimal vectors to find units in cubic fields (see <cit.> and also <cit.>). Minimal vectors are a key ingredient of the proofs of Theorems <ref> and <ref>, they allow to
convert statements about best simultaneous Diophantine approximations of vectors or 
matrices into statements about lattices.


Let M∈ (d+c,ℝ) and let Λ=Mℤ^d+c∈ℒ_d+c. A nonzero vector X∈Λ is a minimal vector in
Λ (with respect to the norms ‖·‖
_ℝ^d and ‖·‖ _ℝ^c) if the
only nonzero vectors Y∈Λ in the cylinder C(X) are such that
C(X)=C(Y), i.e., such that

    | X| _+=| Y| _+, |
    X| _-=| Y| _-.

Two minimal vectors X and Y are equivalent if they define the same cylinder.


 
The connection between minimal vectors and shortest vectors of a lattice in ℝ^d+c with respect to the norm ‖.‖_d,c is simple: in a
given lattice there is at least one shortest vector of the lattice that is minimal, and if X = (u,h)
is minimal in the lattice, then g_tX is a shortest vector of the lattice g_tΛ with t(X) = 1d+cln‖ h‖_ℝ^c/‖ u‖_ℝ^d
(if u=0 or h=0, it is to understand that X is a shortest vector of g_tΛ when t is in a
neighborhood of +∞ or -∞).
Moreover if a vector X = (u,h)  is a “robust” shortest vector, i.e., if
g_tX is a shortest vector in g_tΛ for all t in a neighborhood of t(X), then X is a minimal vector.

 The connection between minimal vectors and shortest vectors of a lattice in ℝ^d+c with respect to the standard Euclidean norm is not that simple: in a given lattice, if g_tX is a shortest vector in the lattice g_tΛ for some t,  then X is a minimal vector in Λ. However some minimal vectors are not of this shape even when d=c=1, this was already observed by Humbert in 1916 (see <cit.> and <cit.>).


The connection between minimal vectors and best Diophantine approximations is given by: 

(<cit.>) 

For all θ in _d,c(ℝ), a vector X=M_θV^T where V=(P,Q)∈ℤ^d+c, is a minimal vector   with nonzero
height in the lattice Λ_θ iff V is a best approximation vector of θ such that

    ‖θ Q-P‖_ℝ^d<d(0,ℤ^d∖{0}).




Let V=(P,Q) be a non zero vector in ℤ^d+c and X=M_θV^T.
Suppose that V is a best approximation vector of θ
such that ‖θ Q-P‖_ℝ^d<d(0,ℤ^d∖{0}). Let W=(u,h) be
a non zero vector in ℤ^d+c. If Y=M_θW^T is in C(X)
then ‖ h‖_ℝ^c≤‖ Q‖_ℝ^c and ‖θ h-u‖_ℝ^d≤‖θ Q-P‖_ℝ^d. The latter inequality implies
h≠0. Since V is a best approximation vector the two previous
inequalities must be two equalities which shows X is minimal with non zero
height. Conversely suppose that X is a minimal vector in Λ_θ
with non zero height. If a vector W=(h,u) in ℤ^d+c is such that ‖ h‖_ℝ^c < or ≤‖ Q‖_ℝ^c, then by minimality, ‖θ h-u‖_ℝ^d> or ≥‖θ Q-P‖_ℝ^d 
which shows that V is a best approximation vector of θ. Moreover, since X is minimal with nonzero height, ‖θ Q-P‖_ℝ^d<d(0,ℤ^d∖{0}).







  §.§.§ The sequence of minimal vectors
 Given a lattice Λ in ℒ_d+c, we select one minimal vector
in each equivalent class of minimal vectors. We number these vectors in
ascending order of heights. Such a numbering exists because 0 is the only
possible limit point of the set of heights of minimal vectors (see the proof of Lemma <ref>). We obtain a
sequence

    ...X_-n(Λ),..., X_-1(  Λ)  , X_0(Λ
    ), X_1(Λ),...

This sequence might be finite, infinite one-sided or two-sided. The sequence
(| X_n(Λ)| _+)_n is decreasing while
the sequence (| X_n(Λ)| _-)_n is 
increasing. Such a numbering is not unique but two such numberings differ only by a constant. Although this is not essential, we will set such a numbering in the Section <ref> about return
times.

We shall  use the following notations

    q_n(Λ)=|  X_n(Λ)|  _-   and   
    r_n(Λ)=|  X_n(Λ)|  _+.



Lemma <ref> above is very important. It shows that for θ∈_d,c(ℝ), the sequences (q_n(Λ_θ))_n
and (q_n(θ))_n are deduced one another by a shift. Therefore, if one of the two limits

    lim_n→∞1/nln q_n(θ), and lim_n→∞1/nln q_n(Λ
    _θ)

exists, then the other exists and have the same value. The same results holds with the sequences (r_n(θ))_n and (r_n(Λ_θ))_n.



 The classical inequality 
 
    q_n+1r_n≤ 1

 which holds for the one-dimensional continued fraction expansion of a real number, can be extended  to minimal vectors of lattices or to best approximation vectors. This fact is well known but it is worth stating it.



There is a constant C_d,c depending only on c and d such that for all lattice Λ∈ℒ_d+c, all matrices θ∈_d,c  and all integers n, we have 

    q_n+1^c(Λ)r_n^d(Λ)≤ C_d,c and 
    q_n+1^c(θ)r_n^d(θ)≤ C_d,c.




Just use the Minkowski convex body theorem with the cylinder defined by two consecutive minimal vectors or best approximation vectors.



The classical inequality

    q_n+2≥ 2q_n

which holds for the denominators of the one-dimensional continued fraction expansion
of a real number, can be extended to minimal vectors of lattices. This
inequality has already been extended to best simultaneous Diophantine
approximations, see <cit.>, <cit.> and <cit.>. The
extension to minimal vectors of lattices is straightforward.


There is a positive integer constant A=A(d,c) such that for
any Λ in ℒ_d+c and any n∈ℤ, if
X_n(Λ), X_n+1(Λ),....,X_n+A(Λ) exist, then

    q_n+A(Λ)     ≥ 2q_n(Λ),
    
    r_n+A(Λ)     ≤1/2r_n(Λ).





Let A be an integer constant such that if A points (x_1,y_1),...,(x_A,y_A) are in the product  of balls B_ℝ^d(0,r_1)× B_ℝ^c(0,r_2) with r_1,r_2≥ 0 then
there exist two indices i≠ j such that ‖ x_i-x_j‖
_ℝ^d≤1/2r_1 and ‖ y_i-y_j‖
_ℝ^c≤1/2r_2. With this choice of the constant A, if k≥ A is a positive
integer such that q_n+k(Λ)≤ 2q_n(Λ), then there are two
integers 0≤ i<j≤ k such that the vector X_n+j(Λ
)-X_n+i(Λ) satisfies both conditions

    {[   | X_n+j(Λ)-X_n+i(Λ)| _+≤1/2r_n(Λ); | X_n+j(Λ)-X_n+i(Λ)| _-≤1/22q_n
    (Λ) ].

which contradicts the definition of X_n(Λ). The same way of
reasoning leads to the other inequality.


We shall use several times the following very simple lemma which is a
consequence of the following observation. For any minimal vector X of a
lattice Λ in ℝ^d+c and any t∈, g_tX is a minimal vector of the
lattice g_tΛ. It follows that


Let Λ be in ℒ_d+c and let t be
in ℝ. The  sequence of minimal vectors of the lattice g_tΛ is  (g_t
(X_n(Λ)))_n.






§ THE TRANSVERSALS S AND S^'


We assume that ℝ^d+c is endowed with the norm

    (x,y)  _d,c=max{  x
    _ℝ^d,  y  _ℝ^c}.

The main idea of the proofs of Theorems <ref> and <ref>
is to induce the flow g_t on the co-dimension one subset

    S={Λ∈ℒ_d+c:λ_2(Λ)=λ_1(Λ)}

where λ_1(Λ) and λ_2(Λ) are the first two minima of the lattice Λ associated with the above norm. 
The reason for introducing  S is that for a generic lattice Λ, one can use the flow to transform any cylinder C(X,Y) defined by two consecutive minimal vectors X and Y, into a cylinder with equal horizontal and vertical sizes which brings the lattice Λ into S (see Subsection <ref>).
And then, to use the Birkhoff ergodic theorem with the first return map associated with the flow g_t. For
technical reason it is better to slightly restrict the definition of S. For instance the
above set is not a submanifold of ℒ_d+c. It could have some “branching
points” while a slightly smaller set is clearly a submanifold, see Lemma
<ref>. It will be convenient to use two transversals S and
S^' for the proof of Theorem <ref>. These two transversals are
very similar; we state all the results we need for both transversals but we only perform 
the proofs for the first transversal S.



 §.§ Definition of S


The transversal S=S_d,c is the set of lattices Λ in ℒ_d+c such
that there exist two independent vectors v_0^S(Λ) and v_1^S(Λ) in
Λ such that:



  * | v_1^S(Λ)| _+ and |
v_0^S(Λ)| _- are <| v_1^S(Λ)|
_-=| v_0^S(Λ)| _+,


  * the only nonzero points of Λ in the ball B_d,c(0,λ_1(Λ)) are ± v_0^S(Λ) and ± v_1^S(Λ).


Observe that for Λ in S, v_0^S(Λ) and v_1^S(Λ) are
unique up to sign and are consecutive minimal vectors of Λ.

Since v_0^S(Λ) and v_1^S(Λ) are linearly independent
and are both in the ball B_d,c(0,λ_1(Λ
)), S is included in the set

    {Λ∈ℒ_d+c:λ_1(Λ)=λ_2(Λ)}.





 §.§ Definition of S^'


The transversal S^' is the set of lattices in ℒ_d+c such
that there exists a vector w_0^S'(Λ) in Λ such that:



  * the only nonzero points
of Λ in the ball B_d,c(0,λ_1(Λ)) are ± w_0^S'(Λ),


  * the ball B_d,c(0,λ_1(Λ)) is equal to the cylinder C(w_0^S'
(Λ)), i.e., |w_0^S'|_+=|w_0^S'|_-.


Observe that w_0^S'(Λ) is unique up to sign and is a minimal vector of
Λ.

We gather some technical results in the next three subsections. Then,  the subsection <ref> about the visiting times and the subsection <ref>    about two functions defined on S, explain the relations between  the transversal S and the sequences of minimal vectors of a lattice.



 §.§ Lattices bases and minima


We shall need the following results about lattices. Recall that a sub-lattice Γ of a lattice Λ is primitive if Γ_ℝ∩Λ=Γ where Γ_ℝ is the vector subspace generated by Γ.


Suppose ℝ^n is equipped with any norm ‖ .‖.
Let Λ be a lattice in ℝ^n and let v_1, v_2 be two
independent vectors of Λ such that ‖ v_1‖ =λ
_1(Λ) and ‖ v_2‖ =λ_2(Λ). Then
ℤv_1+ℤv_2 is a primitive sub-lattice of Λ
unless 1/2(v_1+v_2)∈Λ and ‖ v_1‖
=‖ v_2‖ =‖1/2(v_1+v_2)‖.



Consider the parallelogram 𝒫 defined by the vectors v_1 and
v_2. Let v be an element of Λ that belongs to the interior of
𝒫. If v is not in the segment joining v_1 and v_2 then
the distance from v to either 0 or v_1+v_2 is of the form
‖ t_1v_1+t_2v_2‖ for some positive real numbers
t_1 and t_2 with t_1+t_2<1. Hence this distance is
≤ t_1‖ v_1‖ +t_2‖ v_2‖
<λ_2(Λ) which contradicts the definition of λ
_2(Λ). If v is in the segment joining v_1 and v_2 but
is not the point 1/2(v_1+v_2) then the distance from v to
either v_1 or v_2 is of the form ‖ t(v_1-v_2)‖ with t<1/2 which implies that this distance is <λ
_2(Λ), again a contradiction. If v=1/2(v_1+v_2), we
have ‖ v‖≤1/2(‖ v_1‖
+‖ v_2‖ ) which is <λ_2(Λ) unless
‖ v_1‖ =‖ v_2‖=‖ v ‖ .



It follows that when the norm is strictly convex, the sub-lattice ℤ
v_1+ℤv_2 is always primitive. In our setting despite that the
norm is not strictly convex it is possible to use the above lemma. With our
choice of the norm on ℝ^d+c, the triangle inequality is strict
for two vectors one inside the “top” of the cylinder B_d,c(0,r) and one inside the lateral side of B_d,c(0,r). Therefore,


Let Λ be in S. Then the vectors v_0^S(Λ)
and v_1^S(Λ) associated with Λ are the first two vectors of a
basis of Λ.






 §.§ Geometric properties of S and S^'



S and S^' are submanifolds of
ℒ_d+c of dimension (d+c)^2-2, transverse to the diagonal
flow g_t.



Let Λ_0 be in S and call v_0^S(Λ_0) and v_1^S(Λ_0)
the two vectors provided by the definition of S. By Corollary
<ref>,  v_0^S(Λ_0) and v_1^S(Λ_0)
are the first two vectors of a basis (b_1,...,b_d+c) of Λ_0.
We can find a small enough positive real number ε such that for
any (v_1,...,v_d+c) in the open set

    W=B_ℝ^d+c(b_1,ε)×...× B_ℝ^d+c(b_d+c,ε),



  * the matrix M=M(v_1,...,v_d+c), the columns of which are the v_i, is in (d+c,ℝ) and the sets WP, P∈ (d+c,ℤ) are disjoint,

  * the vectors ± v_1 and ± v_2 are the only nonzero vectors of
the lattice Λ=Mℤ^d+c in the cylinder C(v_1,v_2),

  * ‖ v‖_d,c >‖ v_1‖_d,c and ‖
v_2‖_d,c for all v in Λ∖{0,± v_1,± v_2},

  * |v_1|_+>|v_1|_- and |v_2|_->|v_2|_+.

Consider the map

    f       :W→ℝ^2
         :M=(v_1,...,v_d+c)→(f_1(M)= M,f_2(M)=|
    v_1| _+^2-| v_2| _-^2).

Then a lattice Λ=Mℤ^d+1 with M∈ W, is in S iff
f(M)=(1,0). To prove that S is a submanifold, it is enough to show that the differential
Df(M) is onto at every point M in W. The differential of f_2 is
given by

    Df_2(M).(w_1,...,w_d+c)=2v_1^+.w_1^+-2v_2^-.w_2^-.

The linear map Df_2(M) depends only on w_1 and w_2 and since
v_1^+≠ 0 for all M in W, Df_2(M) is never the zero map. The
differential of f_1 is given by

    Df_1(M).(w_1,...,w_d+c)=∑_i,j=1^d+c(-1)^i+jΔ_i,jw_i,j

where w_j=(w_i,j)_i=1,...,d+c, j=1,...,d+c and Δ_i,j is the
(i,j)-minor of the matrix M. Since M≠ 0, one at least of the
minors Δ_i,3, i≤ d+c, is not zero. Therefore the linear
Df_1(M) is not zero and depends on w_3. It follows that the two linear
maps Df_1(M) and Df_2(M) are linearly independent for all M in W
which implies that S is a submanifold of ℒ_d+c.
To show that the flow is transverse to S, we have to check that for a matrix
M=M(v_1,…,v_d+c) in W such that f(M)=0 we have Df(M).(w_1,...,w_d+c)≠ 0 when
w_i=(cv_i^+,-dv_i^-). Now, for such w_i, Df_2
(M).(w_1,...,w_d+c)=2c| v_1| _+^2 +2d|
v_2| _-^2>0, hence Df(M).(w_1,...,w_d+c) is not zero.





 §.§ Negligible sets

An important ingredient of the proof of Theorem <ref> is that, for a given lattice Λ, the visiting times t, i.e. the times t such that g_tΛ∈ S, can be read from the sequence (X_n(Λ))_n of minimal vectors.
However, this reading is straightforward only for generic lattices, a small subset of lattices has to be avoided.


  §.§.§ A negligible set 𝒩 in the space of lattices 


Let 𝒩=𝒩_d,c be the set of lattices Λ in
ℒ_d+c such that either 


  * there exist two vectors v_1, and v_2, such that
v_1≠± v_2 and | v_1| _+=|
v_2| _+>0 or | v_1| _-=|
v_2| _->0,

  * or there exists a nonzero vector in Λ lying in
the vertical subspace {0}×ℝ^c, 
or in the horizontal subspace
ℝ^d×{0}.



Any  lattice Λ_θ=M_θℤ^d+c  (see Section <ref>) is in 𝒩 because it contains ℤ^d×{0}.


Recall that we have fixed a measure μ  on ℒ_d+c  invariant by the (d+c,ℝ) action. 


𝒩 is μ-negligible and g_t invariant.



Clearly 𝒩 is g_t invariant and the set of lattices with a
nonzero vector in the vertical subspace or in the horizontal subspace is
negligible. So we are reduced to prove that if X≠± Y are two nonzero
vectors in ℤ^d+c the set of matrices M in (d+c,ℝ) satisfying one of the equations

    | MX| _+^2-| MY| _+^2=0

or

    | MX| _-^2-| MY| _-^2=0

is of zero measure. Firstly, by symmetry, it is enough to deal with one of the
equations, say the first one. Secondly, by homogeneity it is equivalent to prove
that the set of matrices in M_d+c(ℝ) that satisfy this equation
is of zero measure. Since this is an algebraic equation, it is enough to prove
that there exists at least one matrix M such that | MX|
_+^2-| MY| _+^2≠ 0. If X and Y are
proportional just choose a matrix M such that |MX|_+≠ 0. Otherwise,
first choose a vector Z in the subspace spanned by X and Y that is
orthogonal to X. Observe that Z.Y≠ 0. Next choose a d-dimensional
subspace V of ℝ^d+c containing Z and orthogonal to X. A
matrix M, the first d rows of which are a basis of V, is such that
| MX| _+=0 and | MY| _+≠ 0.




A lattice Λ that is not in 𝒩 has a
bi-infinite sequence of minimal vectors and is in S iff λ_1
(Λ)=λ_2(Λ). In fact if u and v are two linearly independent shortest vectors in Λ then |u|_+≠|v|_+ and |u|_-≠ |v|_-. Therefore Λ∈ S.




  §.§.§ A negligible set ℳ in the space of matrices _d,c(ℝ)


Let C be a positive real constant and let ℳ=ℳ_d,c=ℳ_d,c(C) be the set of matrices θ∈_d,c()
 such that either 
 
 
  * there exist two nonzero vectors X≠± Y in ^d+c with nonzero heights such that | M_θX| _+=| M_θY| _+
 
  * or there exist infinitely many pairs X≠± Y in ^d+c  such that | X| _-=| Y| _-≠ 0 and 
 | M_θX| _+, | M_θY| _+≤ C| X | _-^-c/d. 
 
 Observe that the second item only occurs when c>1.
 The set ℳ depends on the constant C. Actually, we will only use the value  C=C_d,c where C_d,c is given by Lemma <ref>.

  
  ℳ is negligible with respect to the Lebesgue measure on _d,c(ℝ).
  

  We prove that ℳ is included in a countable union of negligible sets.
 
Given X≠± Y two nonzero vectors in ^d+c with nonzero heights, consider the set ℳ(X,Y) of matrices θ∈_d,c()  such that
 | M_θX| _+=| M_θY| _+. In order to show that ℳ(X,Y) has zero measure it is enough to show that the polynomial 

    f(θ)   =| M_θX| _+^2-| M_θY| _+^2
       =‖ X_+‖ _^d^2-‖  Y_+‖_^d^2
    -2(X_+.θ X_--Y_+.θ Y_-)
     +‖θ X_-‖ _^d^2-‖θ Y_-‖_^d^2

 is not the zero polynomial.

 
If X_-≠± Y_-, we can choose θ_0 such that ‖θ_0 X_-‖ _^d^2-‖θ_0 Y_-‖_^2 ^2≠ 0. With this choice, the one variable polynomial P(t)= f(tθ_0) has a nonzero degree two monomial which implies that the polynomial f is not the zero polynomial. 

If X_-=Y_- (the case X_-=-Y_- is similar), f(θ)=|
X|_+^2-| Y|_+^2-2(X_+-Y_+).θ X_-. Since X_-≠
0, the map φ:θ∈_d,c()→θ X_-∈^d is onto. It follows that we can
choose θ such that θ X_-=X_+-Y_+. With this value of θ, we obtain f(θ)-f(-θ)=-4| X_+-Y_+|^2 ≠ 0 which implies that the
polynomial f is not the zero polynomial.  It follows that ℳ(X,Y) is negligible.


Consider now, for a positive integer n, the set ℳ_n of matrices θ∈_d,c()  such that there is a pair of linearly independent vectors (X, Y)  in ^d+c×^d+c  such 
that n≤| X| _-, | Y| _-<n+1 and 

    | M_θX| _+, | M_θY| _+≤ C|
    X | _-^-c/d.

We want to prove that the set of matrices θ
that are in infinitely many ℳ_n is negligible. We can move into the space _d,c(/) and consider instead  the set 𝒯_n of θ∈_d,c(/) such that there exist q,q' ∈^c linearly independent with n≤‖ q ‖_^c, ‖ q' ‖_^c<n+1 and 

    d(θ q,^d), d(θ q',^d)≤ Cn^-c/d.

For q fixed, the measure of the set of θ∈_d,c(/) such that
d(θ q,^d)≤ Cn^-c/d is a_d,c n^-c where the constant a_d,c depends only on C and the dimensions d and c. When the inequality holds simultaneously for two linearly independent integer vectors q and q', the measure is bounded above by the square of a_d,c n^-c. 
It follows that the measure of 𝒯_n is bounded above by 

    u_n=card{(q,q')∈^c×^c:n≤‖ q ‖_^c, ‖ q' ‖_^c<n+1}× a_d,c^2 n^-2c.

By Borel-Cantelli, it is enough to prove that Σ_n u_n<∞. Now, card{q∈^c:n≤‖ q ‖_^c<n+1}≪ n^c-1, hence

    u_n≪ n^-2

and we are done.





 


 §.§ Visiting times and return maps



Let Λ be in ℒ_d+c. By definition of S, when
g_tΛ is in S, v_0^S(g_tΛ) and v_1^S(g_tΛ)
are two consecutive minimal vectors of g_tΛ. Therefore, 

    {[   v_0^S(g_tΛ)=g_tX_k(Λ); v_1^S(g_tΛ)=g_tX_k+1(Λ) ].

for an integer k.
Hence e^ct| X_k(Λ)| _+=e^-dt|
X_k+1(Λ)| _- which implies

    t=1/d+clnq_k+1(Λ)/r_k(Λ).

It follows that the set of real numbers t such that g_tΛ∈ S is
included in the discrete set

    V_Λ(S)={t_k=1/d+clnq_k+1(Λ)/r_k(Λ
    ): k∈ℤ}.

It can happen that some values t_k are skipped, but in that case, Λ
must be in 𝒩. So, when Λ is not in 𝒩,
g_tΛ∈ S iff t∈ V_Λ(S). For the transversal S^', the
same results hold with

    V_Λ(S')={t_k^'=1/d+clnq_k(Λ
    )/r_k(Λ): k∈ℤ}.

It follows that for almost all Λ, both the backward trajectory
(g_tΛ)_t≤ 0 and the forward trajectory (g_tΛ)_t≥ 0
visit the two tranversals  S and S^'  infinitely often. 
Therefore the first return/entrance times in
S and S^',

    τ(Λ)       =inf{t>0:g_t(Λ)∈ S}∈ℝ_>0∪{∞},
    τ^'(Λ)       =inf{t>0:g_t(Λ)∈ S^'}∈ℝ_>0∪{∞},

are finite almost everywhere and 
the first return/entrance maps 

    R(Λ)       =g_τ(Λ)Λ,
    
    R^'(Λ)       =g_τ^'(Λ)Λ

are defined for all Λ that are not in 𝒩.

For an integer n≥ 1, let us denote τ_n the n-th return (or entrance)
time in S, i.e.

    τ_n(Λ)=∑_k=0^n-1τ(R^k(Λ))

(R^0(Λ)=Λ for all Λ in ℒ_d+1).
It will be convenient to choose the numbering of the sequence of minimal vectors (X_n(Λ)) in order to have simple formulas for the return time and the return map.



Numbering convention: For a lattice Λ∈ℒ_d+c, n=0
is the smallest integer n∈ℤ such that

    | X_n+1(Λ)| _-≥| X_n(Λ
    )| _+
 
when the set of such integers is non empty.


With this numbering convention, for all Λ in S, we have 

    X_0(Λ)=v_0^S(Λ)

and for all Λ in S', we have 

    X_0(Λ)=w_0^S'(Λ).

Moreover when Λ∉ S is not in 𝒩,

    | X_1(Λ)| _--| X_0(Λ
    )| _+> 0,
       τ(Λ)=1/d+cln(| X_1(Λ)|_-/| X_0(Λ
    )|_+),

and

    g_τ(Λ)X_0(Λ)=± v_0^S(R(Λ))=± X_0(R(Λ)).




Let us summarize the above.


Let Λ be a lattice in ℒ_d+c.1. The set of visiting times in S is included in V_Λ(S)={t_k=1/d+clnq_k+1(Λ)/r_k(Λ
): k∈ℤ} and the set of visiting times in S' is included in V_Λ(S')={t_k^'=1/d+clnq_k(Λ
)/r_k(Λ): k∈ℤ}. 2. Suppose Λ is not in 𝒩. The set of visiting times in S is equals to V_Λ(S) and the set of visiting times in S' is equals to V_Λ(S').




For θ  in _d,c(), we also need to connect the visiting times of the transversal S with the best approximation vectors of θ. 



Let θ  be in _d,c()∖ℳ. Then for all large enough integers n, 


    t_n(θ)
    =1/d+clnq_n+1(θ)/r_n(θ)
 
and 

    t'_n(θ)
    =1/d+clnq_n(θ)/r_n(θ)

are visiting times for the transversals S and S' respectively.

 


Let θ  be in _d,c()∖ℳ. Consider the sequence of all best approximation vectors (Y_n(θ))_n∈ of θ. By Lemma <ref>, there are integers n_1 and k such that X_n+k(Λ_θ)=M_θY_n(θ) for all n≥ n_1. Since θ is not in ℳ, by Lemma <ref> there is another integer n_2 such that for all n≥ n_2, the only nonzero vector of Λ_θ in the box 𝒞(X_n+k(θ),X_n+k+1(θ)) are ± X_n+k(θ) and ± X_n+k+1(θ). This means that for all n large enough, the times 

    t_n(θ)=1/d+clnq_n+1(θ)/r_n(θ) and 
    t'_n(θ))
    =1/d+clnq_n(θ)/r_n(θ)
 
are  visting times for the transversals S and S'.




 §.§ Functions defined on S


Let Λ be in S. By definition of S, the functions

    ρ,ρ^∗:Λ∈ S→ln| v_1^S(Λ)| _-/| v_0^S(Λ)| _-
    , ln| v_0^S(Λ)| _+/|
    v_1^S(Λ)| _+∈ℝ_>0∪{+∞}

are well defined on S. The next lemma is easy, its proof is close to the beginning of
the proof of Lemma <ref> and is omitted.


The functions Λ∈𝒮→  |v_0^S(Λ)|_-, |v_0^S(Λ)|_+, |v_1^S(Λ)|_-, |v_1^S(Λ)|_+ are continuous and thus the functions ρ and ρ^∗ are continuous.


The following lemma is important. On the one hand, it will imply that the functions ρ and ρ^∗ are integrable. On the other hand, it will explain the connection between the Lévy's constant
L_d,c and the average return times on S.


Let Λ be a lattice in S∖𝒩.
Then

    τ(Λ)=1/d+c(ρ(R(Λ))+ρ^∗(Λ)).





Let Λ be in S∖𝒩. By definition of S,
q_1(Λ)=r_0(Λ). Hence, by Lemma <ref>,

    (d+c)τ(Λ)       =lnq_2(Λ)/r_1(Λ)×r_0(Λ)/q_1(Λ)
         =lnq_2(Λ)/q_1(Λ)+lnr_0(Λ)/r_1(Λ)
         =ρ(R(Λ))+ρ^∗(Λ).






§ INDUCED MEASURES ON S AND S^'



 §.§ Definition of the induced measures μ_S and μ_S'


Recall that we have fixed a measure μ  on ℒ_d+c  invariant by the action of (d+c,ℝ). Let us explain how the flow g_t and the measure μ induce a measure μ_S on S which is invariant by the first return map R. The construction of such a measure is standard but we have not found any reference suitable for our case. 

Let X=(S∖𝒩)×, let π:X→ℒ_d+c∖𝒩 be the map defined by π(Λ,t)=g_tΛ, let g_t^X:X→ X be the flow defined by g_t^X(Λ,s)=(Λ,s+t), let R:X→ X be the map defined by R(Λ,t)=(R(Λ),t-τ(Λ)) and let D={(Λ,t)∈ X:0≤ t<τ(Λ)}.


  * Since the first return map R is a bijection from S∖𝒩 onto itself, R is a bijection with inverse R^-1(Λ,t)=(R^-1(Λ),t+τ(R^-1(Λ)). It is easy to check that the sets R^n(D), n∈, are disjoint.
Since the infinite sequence τ_n(Λ) strictly increases from -∞ to +∞, X=∪_n∈R^n(D). 

  * We can lift the measure μ from ℒ_d+c to X: we define the  measure μ on X by 

    μ(B)=∑_n∈μ(π(D∩R^ -n B))

for all Borel sets B in X. By definition μ is R-invariant.

  * Let us see that μ is  g_t^X-invariant for all t∈. If B is a Borel set in X, we have the partition B=∪_m,n∈B_m,n where B_m,n=B∩R^mD∩ g_-t^X R^nD. 
On the one hand, since B_m,n⊂R^mD, we have μ(B_m,n)=μ(π(D∩R^-mB_m,n))=μ(πR^-mB_m,n), hence μ(B_m,n)=μ(π B_m,n)
because π∘R=π. On the other hand, since g_t^XB_m,n⊂R^nD,

    μ(g_t^XB_m,n)=μ(R^-ng_t^XB_m,n)=μ(πR^-ng_t^XB_m,n)=μ(π g_t^XR^-nB_m,n)=μ(g_tπR^-nB_m,n),

and since μ is g_t-invariant, we obtain, μ(g_t^XB_m,n)=μ(B_m,n).

  * Since μ is g_t^X-invariant, there exists a measure μ_S on S∖𝒩 such that

    μ=μ_S⊗Lebesgue_.


  * By definition for any Borel set B⊂ S and any δ>0 such that B⊂{Λ:τ(Λ)>δ},

    μ_S(B)×δ=μ(B×[0,δ])=μ(∪_ t∈[0,δ]g_tB).


  * Let us show that μ_S is R-invariant. Let B be a Borel set in S∖𝒩. Since μ is R-invariant,

    μ_S(B)   =μ(B×[0,1])=μ(R(B×[0,1]))
       =∫_R(B)dμ_S(Λ)∫_-τ(R^-1Λ)^1-τ(R^-1Λ)dt=μ_S(R(B)).


  * We extend μ_S to S by μ_S(S∩𝒩)=0.


With our definition of μ_S, and since π(D)=ℒ_d,c∖𝒩, the Kac return-time theorem reduces to

    μ(ℒ_d+c)=μ(π(D))=μ(D)=∫_Sτ(Λ)dμ_S(Λ).


Finally, the measure μ_S is R-ergodic because any R-invariant set of positive measure in S gives rise to a set of  flow trajectories of positive measure and it is known that the action of the flow (g_t) on (ℒ_d+c,μ) is ergodic (see <cit.>).

The flow induces a measure μ_S' on S' as well. 


 §.§ Finiteness of the induced measures
 
Let us prove that the induced measures μ_S and μ_S' are finite. This is a simple consequence of the next lemma which will be very important in the proof of Theorem 
<ref> about the almost sure convergence in M_d,c(ℝ).


 1. There exists an integer constant A such that
τ_A(Λ)≥ 1 for all Λ in S
 and  τ_A^'(Λ)≥ 1 for all Λ in S'.



Let A be the constant given by Lemma <ref> about the growth rate of the sequences (q_n(Λ))_n and (r_n(Λ))_n. For all integers k, we have 

    1/d+c(lnq_k+A+1(Λ)/r_k+A(Λ)-
    lnq_k+1(Λ)/r_k(Λ))
       ≥1/d+clnq_k+A+1(Λ)/q_k+1(Λ)
       ≥1/d+cln 2.

Since by Lemma <ref>, the set of visiting times of Λ is included in

    V_Λ(S)={t_k=1/d+clnq_k+1(Λ)/r_k(Λ
    ): k∈ℤ},

τ_A(Λ)≥1/d+cln 2. Multiplying A by the smallest integer larger than d+c/ln 2 we are done.




μ_S(S) and μ_S^'(S^') are finite and nonzero.



By Lemma <ref>, S is submanifold of co-dimension one, transverse to the flow, and nonempty, hence  μ_S(S)>0.

Since μ_S is R-invariant,  for all k

    ∫_Sτ(R^kΛ)dμ_S(Λ)=∫_Sτ(Λ)dμ
    _S(Λ),

and by the Kac return time theorem,

    ∫_Sτ(Λ)dμ
    _S(Λ)=μ(ℒ_d+1),

therefore,

    ∫_Sτ_A(Λ)dμ_S(Λ)=∫_S∑_k=0^A-1τ
    (R^kΛ)dμ_S(Λ)=Aμ(ℒ_d+c).

By the above lemma, τ_A≥ 1 on S, hence μ_S(S)≤ A μ(ℒ_d+c) which is finite by Siegel's theorem.





§ ALMOST SURE CONVERGENCE IN THE SPACE OF LATTICES




 §.§ Consequence of  the Birkhoff ergodic theorem

Recall that for a lattice Λ∈ℒ_d+c, q_n(Λ)=|  X_n(Λ)|  _- and r_n(Λ)=|  X_n(Λ)|  _+ where (X_n(Λ))_n is the sequence of minimal vectors of Λ with the numbering convention of Section <ref>. Recall also that τ_n(Λ) is the n-th return time in S.

There exist two positive constants L_d,c and
L_d,c^∗ such that for almost all lattices Λ in ℒ_d+c,

    lim_n→∞1/nln q_n(Λ)       =1/μ
    _S(S)∫_Sρ dμ_S=L_d,c>0,
    lim_n→∞-1/nln r_n(Λ)       =1/μ
    _S(S)∫_Sρ^∗dμ_S=L_d,c^∗>0,
    lim_n→∞1/nτ_n(Λ)       =1/d+c(L_d,c+L_d,c^∗)=μ(ℒ_d+c)/μ_S(S).

Moreover, given two new  Euclidean norms on the horizontal space ℝ^d×{0} and the vertical space ℝ^c×{0},  if we define the minimal vectors and the transversal S with these two new Euclidean norms, then the conclusion of the theorem remains valid with the same constant L_d,c and L^*_d,c.


The above theorem is an important result in the direction of Theorem <ref>.  The big difference between the two theorems is the reference measure. However, the second statement in Theorem <ref> explains why Theorem <ref> remains valid with the same constant L_d,c and L^*_d,c for any pair of Euclidean norms.  


Let Λ be in S∖𝒩. By Lemma <ref>,
τ(Λ)=1/d+c(ρ(R(Λ))+ρ^∗(Λ)). Because
the space of lattices has finite measure, the return time τ is in
ℒ^1(S) and since  the non negative functions ρ∘ R
and ρ^∗  are bounded above by (d+c)τ, they are also in ℒ^1(S). Making use of the
Birkhoff ergodic theorem with the functions ρ and ρ^∗, we obtain the
almost everywhere convergence of the sums

    1/N∑_n=0^N-1ρ∘ R^n, 1/N∑_n=0^N-1ρ^∗∘ R^n

on S to R-invariant functions. Now the ergodicity of the flow g_t (see <cit.>)
implies the ergodicity of the return map R. Therefore 1/N∑_k=0^N-1ρ∘
R^n and 1/N∑_n=0^N-1ρ^∗∘ R^n converge almost
everywhere on S to the constants 

    L_d,c=1/μ_S(S)∫_Sρ dμ_S and 
    L_d,c^∗=1/μ_S(S)∫_Sρ^∗dμ_S.

We would like to see that the Birkhoff sums converge to the same limits almost everywhere in the whole space of lattices. Let Λ be in
ℒ_d+c∖(S∪𝒩) and let n be  a positive integer. By
Lemma <ref> and the numbering convention, for all n≥ 1,

    ρ∘ R^n(Λ)       =ln| X_1(R^n(Λ
    ))| _-/| X_0(R^n(Λ))| _-
         =ln| g_τ_n(Λ)(X_n(Λ))|
    _-/| g_τ_n(Λ)(X_n-1(Λ))| _-
         =ln| X_n(Λ)| _-/|
    X_n-1(Λ)| _-

and ρ^∗∘ R^n(Λ)    =ln| X_0(R^n
(Λ))| _+/| X_1(R^n(Λ))| _+
  =ln| X_n-1(Λ)| _+/|
X_n(Λ)| _+
as well. It follows that if the Birkhoff sums 1/N∑_n=1^Nρ∘ R^n(Λ) and 1/N∑_n=1^Nρ^∗∘
R^n(Λ) converge to L_d,c and
L_d,c^∗, then

    lim_N→∞1/Nln q_N(Λ)    =L_d,c and lim_N→∞-1/Nln r_N(Λ)   =L_d,c^∗.

Now the image by the map R:ℒ_d+c→ S of a subset of
nonzero measure in ℒ_d+c is a set of nonzero measure in S, therefore the sums 1/N∑_n=1^Nρ∘ R^n and 1/N∑_n=1^Nρ^∗∘ R^n converge almost everywhere in ℒ_d+c to
L_d,c and L_d,c^∗.

By Lemma <ref>, we know that the sequences (q_n(Λ))_n
and (r_n(Λ)^-1)_n have at least exponential growth rate;
therefore, the constants L_d,c and L_d,c^∗ are >0.


By Lemma <ref>, for all Λ in S∖𝒩
and k∈ℕ,

    τ_k+1(Λ)-τ_k(Λ)=1/d+c(ρ(R^k+1
    (Λ))+ρ^∗(R^k(Λ))),

hence

    lim_n→∞1/nτ_n(Λ)=1/d+c(L_d,c+L_d,c^∗)

almost everywhere.

Finally, let us proof that the constants L_d,c and L_d,c^∗
do not depend on the Euclidean norm on ^d and ^c.  For a matrix M_d in 
(d,) and a matrix M_c in 
(c,),
let us denote M the matrix

    M=(
    [ M_d   0;   0 M_c ]) ∈(d+c,ℝ) .

Since the action of M on
ℒ_d+c is measure preserving,

    lim_n→∞1/nln q_n(AΛ)       =L_d,c
    lim_n→∞-1/nln r_n(AΛ)       =L_d,c^∗

almost everywhere in ℒ_d+c. Now, a vector AX in the lattice
AΛ is  minimal  iff X is a minimal vector of the  lattice Λ with respect to
the new Euclidean norms ‖ .‖ _A_d,ℝ^d  and ‖ .‖ _A_c,ℝ^c where

    ‖ u‖ _M_d,ℝ^d=‖ M_du‖
    _ℝ^d,  and ‖ v‖ _M_c,ℝ^c=‖ M_cv‖
    _ℝ^c.

Since up to  multiplicative constants, all the Euclidean norms are of the above form, the constants L_d,c and L_d,c^∗ do not depend on the Euclidean norms on ℝ^d and ^c.





 §.§ A consequence of the Borel-Cantelli lemma

For any lattice Λ∈ℒ_d,c∖𝒩 and any integer n, we have q_n+1(Λ)r_n(Λ)=λ_1(g_t_nΛ)λ_2(g_t_nΛ)
where t_n=τ_n(Λ)=1d+clnq_n+1(Λ)/r_n(Λ) is the n-th return time in S.   
When d=c=1,   λ_1(g_t_nΛ)λ_2(g_t_nΛ)≍ 1 by the Minkowski second convex body theorem. It implies that L_1,1=L^*_1,1. When d or c≥ 2, the product of the first two minima of a lattice in ℒ_d,c can be arbitrarily small, so it is not clear whether there is a relation between L_d,c and L^*_d,c. However,


With the notation of Theorem <ref>, we have

    cL_d,c=dL_d,c^∗.




The function Λ∈ℒ_d+c→ 1/λ_1(Λ) is in L^1(μ).


By Siegel formula (see <cit.>) used with the function 
f=1_B_^d+c(0,r), we have for some positive constants C_1 and C_2 depending on the dimension d+c and on the normalization of the Haar measure, 

    C_1r^d+c=∫_^d+cf(x)dx=C_2∫_ℒ_d+c∑_x∈Λ∖{0}f(x)dμ(Λ)
    ≥ C_2μ({Λ:λ_1(Λ)≤ r }).
 
Therefore ∫_ℒ_d+c1/λ_1dμ=∫_0^∞μ({Λ:1/λ_1(Λ)≥ t })dt≤μ({ℒ_d+c}) +C_1/C_2∫_1^∞1/t^d+cdt<∞.


The inequality cL_d,c≤ dL_d,c^∗ is easy to prove. By Lemma <ref>, for all lattices Λ in ℒ_d+c and all n, we have
q_n+1^c(Λ)r_n^d(Λ)≤ C_d,c. Hence for a lattice Λ
such that

    lim_n→∞1/nln q_n(Λ)    =L_d,c and lim_n→∞-1/nln r_n(Λ)    =L_d,c^∗,

we have,

    cL_d,c-dL_d,c^∗=clim_n→∞ln q_n(Λ)/n+dlim_n→∞ln r_n(Λ)/n=lim_n→∞ln q_n^c(Λ)r_n^d(Λ)/n≤ 0.

The converse inequality uses the Borel-Cantelli lemma. Let φ:]0,∞
[→]0,∞[ be a decreasing function such that
∑_n≥ 1φ(n)<∞, for instance φ(t)=1/t^α with α>1. Since for such a function φ, liminf
_n→∞1/nlnφ(n)=0, the inequality
cL_d-dL_d^∗≥ 0 holds, provided that for almost all lattices
Λ, we have q_n^c(Λ)r_n^d(Λ)≥φ(n)^d+c
for n large enough.
Let K be a constant that will be chosen later. For each integer n≥ 1,
consider the set A_n of lattices Λ in ℒ_d+c such
that

    λ_1(Λ)≤ Kφ(n)

and the set B_n=g_t_nA_n where t_n=1/d(lnφ(n)-n).
By the above lemma ‖λ_1^-1‖ _1<∞, hence using  Markov inequality, we obtain

    μ(B_n)=μ(A_n)≤‖λ_1^-1‖ _1/1/Kφ(n)≪φ(n).

Therefore, by the Borel-Cantelli lemma, the set ℬ of lattices
Λ in ℒ_d+c such that Λ∈ B_n for infinitely
many integers n, is negligible.
Suppose now that Λ is a lattice such that q_n^c(Λ)r_n
^d(Λ)≤φ(n)^d+c for infinitely many n. For each integer
n≥ 1, let k_n=k_n(Λ)= ⌊ln q_n(Λ)⌋. By
Theorem <ref>, for almost all lattices, we have k_n≤
(L_d,c+1)n for n large enough. Therefore, for almost all lattices Λ, for n
large enough, if q_n^c(Λ)r_n^d(Λ)≤φ(n)^d+c
then the vector g_t_k_n^-1(X_n(Λ)) satisfies both

    | g_t_k_n^-1(X_n(Λ))| _+   
    =r_n(Λ)e^c/d(k_n-lnφ(k_n))
       ≤ r_n(Λ)q_n^c/d(Λ)φ(k_n)^-c/d
       ≤φ(n)^d+c/dφ(k_n)^-c/d
         =φ(k_n)×(  φ(n)/φ(k_n))
    ^d+c/d
       ≤φ(k_n)×(  φ(n)/φ((L_d,c
    +1)n))  ^d+c/d≤ Kφ(k_n)

for some constant K depending only on φ (we use that φ(t)/φ((L_d,c+1)t) is bounded above which is obviously true
when φ(t)=1/t^α) and

    | g_t_k_n^-1(X_n(Λ))| _-=q_n
    (Λ)e^-k_n+lnφ(k_n)≤ eφ(k_n)≤ Kφ
    (k_n).

Thus there are infinitely many n such that Λ∈ B_k_n. Since the
sequence (k_n)_n goes to infinity, Λ∈ℬ. It follows
that for almost all lattices Λ, q_n^c(Λ)r_n^d
(Λ)≥φ(n) for n large enough and we are done.



As an immediate consequence of the previous proposition and of Theorem <ref>, we have



    L_d,c=d/μ_S(S)∫_Sτ dμ_S=d×μ
    (ℒ_d+c)/μ_S(S).








§ PARAMETRIZATION OF S WHEN C=1 



This section is not necessary neither for the proofs of Theorems
<ref> and <ref>, nor for Sections <ref> and <ref>.
The aim is to show that the computation of the constant L_d,c is theoretically feasible in the case c=1. However, if the case d=1 is easy (see below), the case d=2 is already difficult. It is possible to give an integral formula for L_2,1. But, we are not able to compute the integral, only a numerical estimation of the integral has been carried out. 
An exact description of S when d≥ 3 seems to be rather difficult.
In this section we assume c=1.


 §.§ rkN decomposition 


In this subsection we give a parametrization of a set of lattices that contains S.

Let Λ be a lattice in S and let u_1=v_0^S(Λ) and
u_2=v_1^S(Λ) be the two vectors associated with Λ by the
definition of S (see the definition of S in Section <ref>). When d≥ 2, we suppose these two
vectors have non negative heights and when d=1, we only suppose that u_2 has a non negative height. Since u_1 and u_2 are independent shortest vectors, by Corollary <ref>, they
 are the first two vectors of a basis of Λ. Thus, there is a
matrix M∈(d+1,ℝ) defining Λ the first two columns of
which are the vectors u_1 and u_2. 

When d=1, using the scaling factor r=| u_1|_+=| u_2|_->0, we can write
M =rN,
where N is in the set U_1 of 2× 2 matrices such that

    n_1,1=n_2,2=1  and | n_2,1|, | n_1,2|   <1.


When d≥ 2, let us denote (e_1,e_2,...,e_d+1) the standard basis of ℝ^d+1. Using the same scaling factor 
r=| u_1|_+=| u_2|_->0
and an orthogonal matrix k that fixes e_d+1  and sends e_1 to 1/ru_1,+, we can find a matrix
N=(n_i,j)_1≤ i,j≤ d+1 such that M    =rkN,  N    >0 and

    n_1,1=n_d+1,2=1>n_d+1,1=| u_1|_-≥ 0,
    ‖ (n_1,2,...,n_d,2)‖ _ℝ^d<1,
    
    
    n_2,1=...=n_d,1=0.




When d≥ 2, k is chosen in the group

    K_d={k∈(d+1):ke_d+1=e_d+1}.

and using the decomposition of a d× d matrix in a product of an orthogonal 
matrix with positive determinant and of an upper triangular matrix, we can even suppose that

    n_i,j=0, for all 1≤ j<i≤ d.



For d≥ 2, let us denote U_d the set of (d+1)×(d+1)-matrices such that
(<ref>), (<ref>),  and
(<ref>) hold ((<ref> implies (<ref>)). 

Since M=1, the scaling factor r must be equal to ( N)^-1/d+1.
Puting K_1={I_2}, for all d≥ 1, the map

    (k,N)∈ K_d× U_d→ ( N)^-1/d+1kN

provides a natural parametrization of a subset Σ in
(d+1,ℝ) whose  projection
in ℒ_d+1  contains S. The main problem is now to find
which of these couples (k,N) are such that rkNℤ^d+1∈ S and
to select a fundamental domain in this set of couples. This problem reduces to

	
  * finding the set of matrices N∈ U_d such that the first two columns u_1 and u_2 of N are in the unit ball
	B_d,1(0,1) and are the only nonzero vectors of the lattice
	Nℤ^d+1 in this ball,
	
  * then select a fundamental domain in this set of matrices N.

This is easy when d=1 and doable when d=2. When
d=1, it is even possible to find the first return map R.

Another issue is to find the measure μ_S on S induced by the flow
g_t and the invariant measure μ of ℒ_d+1. This comparatively easier
issue can be performed for all d without knowing explicitly S.



 §.§ The induced measure μ_S


Consider the manifold V_d=ℝ_>0×ℝ×K_d× U_d and the  submanifold

    W={(Δ,t,k,N)∈ V_d: Δ=1, t=0}={1}×{0}×
    K_d× U_d

together with the map F  :V_d→(d+1,ℝ) defined by

    F(Δ,t,k,N)= (Δ/ N)
    ^1/d+1g_tkN

and  the map F:V_d→(d+1,ℝ)/(d+1,ℤ) defined by

    F(Δ,t,k,N)=F(Δ,t,k,N)ℤ^d+1.

By the discussion of the previous subsection, F provides a parametrization of S:
S⊂F(W).
We would like to compute the measure μ_S in the coordinates (1,0,k,N). The submanifold W is equipped with the reference measure 
μ_K_d⊗λ_U_d
where λ_U_d is the Lebesgue measure on U_d and  μ_K_d is the invariant measure on K_d associated with the invariant volume form γ
on K_d that is dual to the exterior product of 
the	invariant vector fields generated by the standard skew symmetric matrices
(A_i,j=E_ij-E_ji)_1≤ j<i≤ d.

We have to precise the normalization of the Haar measure that we use on (d+1,).  Consider the standard volume form ω on _d+1(ℝ) seen as ℝ^(d+1)^2 and the
vector fields X and Y on _d+1(ℝ) defined by

    Y(M)=M  and  X(M)=∂(g_tM)/∂ t|_t=0.

The invariant measure in (d+1,ℝ) is given by the volume form
1/(d+1)^2i_Yω. With this
normalization of the volume form we have the beautiful Siegel formula

    μ(ℒ_d+1)=*vol((d+1,ℝ)/(d+1,ℤ))=1/d+1∏_k=2
    ^d+1ζ(k)

(see <cit.>, p. 152). The induced measure μ_S on S is
given by the restriction to S of the projection in the space of lattices of the differential form

    1/(d+1)^2i_Xi_Yω.

Using this differential form it is possible to compute the induced measure μ_S with the parametrization F, we give without proof an explicit formula in next proposition.
 
 Assume d ≥ 2. Suppose that 𝒟 is an open subset of W such that F(𝒟)⊂ S and the restriction  of F to 𝒟 is one to one. Then the image by F of the measure 

    1_𝒟 (1/ N)^d+1(
    	1/d+1∏_j=2^d-1
    	n_j,j^d-j) μ_K_d⊗λ_U_d

is the restriction of μ_S to F(𝒟).







 §.§ Determination of the transversal S, c=1, d=1 

We already have a map 

    U_1→ℒ_2
        N→ ( N)^-1/2Nℤ^2

that sends U_1 onto a set that contains S.
By definition of S, the image of a matrix

    N=(
    [     1 n_1,2; n_2,1     1 ])∈ U_1

is in S iff the only nonzero vectors of the lattice Λ=Nℤ^2 in the ball B_1,1(0,1) are the two columns of N up to sign.  We obtain that  Λ∈ S iff

	
  * 0<| n_2,1|, | n_1,2| <1,
	
  * the signs of n_1,2 and n_2,1 are opposite.

So the map F defined on ]0,1[^2×{-1,1} defined by 

    (x,y,ε)→1/(1+xy)^1/2(
    [    1 -ε x;  ε y    1 ])ℤ^2

provide a parametrization of S and it is easy to see that F is a bijection. An easy computation of a 4× 4 determinant gives that 

    i_Xi_Y14ω(∂ F∂ x,∂ F∂ y)=14ω(X,Y,∂ F∂ x,∂ F∂ y)=1/2(1+xy)^2,

hence

    f(x,y,ε)=1/2(1+xy)^2
  
is the density the measure μ_S with respect to the Lebesgue measure.
Therefore μ_S(S)=ln 2.
With the Siegel formula (<cit.>, p. 152) and Corollary  <ref>, we obtain the Lévy's constant

    L_1,1=μ(ℒ_d+c)/μ_S(S)=ζ(2)/2ln 2
    =π^2/12ln 2.
 



  §.§.§ Determination of the first return map, c=1, d=1

Let 

    Λ=F(x,y,ε)=1/(1+xy)^1/2(
    [    1 -ε x;  ε y    1 ])ℤ^2

be in S.   By Lemma <ref>, to find the first return map R(Λ) in S, it is enough to find the minimal vector X_2(Λ). Then R(Λ) is given by 
g_τ(λ)(Λ) with 

    τ(Λ)=1/2ln(| X_2(Λ)|_-/| X_1(Λ
    	)|_+).

By corollary <ref>, the first minimal vectors
X_0(Λ) and X_1(Λ) form a basis of Λ. The
minimal vector X_2(Λ) is the vector of the form
X=aX_0(Λ)+bX_1(Λ)  in the strip | X
|_+<| X_1(Λ)|_+=x
with a,b∈ℤ, and with the smallest height. It is not difficult to see that 

    X_2(Λ)=ε X_0(Λ)+⌊1x⌋ X_1(Λ).

So we obtain R(Λ)=F(x',y',ε') where

    ε'=-ε ,  
    x'={1x} and 
    y'=1/y+⌊1x⌋.

It follows that the return map R is a two-fold extension of the natural extension of the Gauss map.



 §.§ Value of Lévy's constant when d=2 and c=1, 

An exact description of S is possible when d=2 and c=1. Together with the expression of the measure μ_S in Proposition <ref>, this leads to a closed formula for Lévy's constant as a seven-tuple integral of an algebraic function over a union of domains the boundaries of which are algebraic surfaces of degree at most two. We are not able to compute this seven-tuple integral. However using Octave, Seraphine Xieu (see <cit.>,<cit.>) has computed a numerical approximation of Levy's constant

    L_2,1=1.135256974…

This can be compared with the one dimensional Levy's constant

    L_1,1=1.186569111…




§ ALMOST SURE CONVERGENCE IN M_D,C(ℝ)

We want to prove that   lim_n→∞1/nln q_n(Λ_θ)=L_d,c for Lebesgue-almost all  θ∈ M_d,c(). By Theorem <ref>, we already know that lim_n→∞1/nln q_n(Λ)=L_d,c for μ-almost all lattices Λ∈ℒ_d+c, but we have to change the  reference measure. Furthermore,   the set ℋ_> of lattices of the shape Λ_θ, θ∈ M_d,c(), is μ-negligible. Actually, we will first prove a general result,  Theorem <ref> (see below): the Birkhoff sums 1/N∑_n=0^N-1φ∘ R^n(Λ_θ) converge for almost all θ when the function φ satisfies some regularity assumptions. The main ingredient of the proof of this general result is that ℋ_> contains the expanding direction of the flow g_t.   However, Theorem <ref> also requires that Lemma <ref> below holds for the transversal. This is proven in Subsections <ref> and <ref> with an example. These two subsections are not necessary for the proof of Theorem <ref>.  



 §.§ A general result


Recall that ℋ_≤ is the subgroup of (d+c,ℝ)
defined by

    ℋ_≤={h∈(d+c,ℝ):h=(
    [ A 0; B C ])  }

with A∈(d,ℝ), B∈_c,d(ℝ) and C∈(c,ℝ). We say that a function f:ℒ_d+c→ℝ is uniformly continuous in the ℋ_≤-direction if for all ε there exists β>0 such that
for all Λ∈ℒ_d+c and all h∈ B_ℋ_≤
(I_d+c,β), | f(hΛ)-f(Λ)|≤ε.


 1. Let φ:S→ℝ be a function
continuous almost everywhere on S. Suppose there exists a non negative function f:ℒ_d+c→ℝ_≥ 0 that is continuous,
uniformly continuous in the ℋ_≤-direction, integrable and
  such that |φ|≤ f on S. Then,

    ∫_Sfdμ_S<+∞

and for almost all θ in _d,c(ℝ),

    lim_n→∞1/n∑_k=0^n-1φ∘ R^k
    (Λ_θ)=1/μ_S(S)∫_Sφ dμ_S.

2. The same result holds for S^' instead of S


We
can formulate Theorem <ref> for  a general transversal S. The assumptions about S are:



  * S is a co–dimension one submanifold transverse to the flow,


  * the number of visiting times in a time interval of length 1 is bounded
above by a universal constant A (Lemma <ref>),


  * Lemma <ref> below holds for S.


The other assumptions and the conclusion are the same as in Theorem <ref>.



For a compact subset K of the submanifold S and δ>0, let us denote

    U(K,δ)={g_thΛ:t∈[0,1], h∈ B_ℋ_≤
    (I_d+c,δ), Λ∈ S∖ K}.




 For all ε>0, there exist a compact subset
K in S and δ>0 such that μ(U(K,δ))≤ε.


 This is an important lemma because it explains that the
part of S near its “boundary” is not
relevant. This lemma  holds  for S' as well but it will be proven below only for S.

We shall prove first a particular case of Theorem <ref> which also requires Lemma <ref>: 





 Let φ:S→ℝ be a bounded
continuous function. Then for almost all θ in _d,c(ℝ),

    lim_n→∞1/n∑_k=0^n-1φ∘ R^k
    (Λ_θ)=1/μ_S(S)∫_Sφ dμ_S.






  §.§.§ Auxiliary lemmas

The proof of Theorem <ref> needs three more lemmas.
The first one only uses  that S and S' are transverse to the flow together with the inverse function theorem. In the remaining of the section <ref>, the balls B(I_d+c,r) are in (d+c,).

	 For all compact subset K in S (or in S'), there exist α and
	η>0 such that 
	- the map (t,Λ)→ g_tΛ is
	one to one on [-α,α]× K,
	- for all h∈
	B(I_d+c,η) and all Λ in K, there exists an unique
	t=t(h,Λ)∈[-α,α] such that g_-thΛ∈ S.
	- the maps σ:(h,Λ)→ t=t(h,Λ) and π
	:(h,Λ)→ g_-thΛ are continuous on B(I_d+c
	,η)× K and the values of σ are in [-α/4,α/4].



Let ψ:(t,Λ)∈× S→ g_tΛ∈ℒ_d+c.	
Since S is a submanifold and since the flow is transverse to S, thanks to the  inverse function theorem, we see that for each Λ∈ S, there exists 	an open neighborhood W_Λ of Λ in S and a_Λ>0 such that ψ is a diffeomorphism from ]-a_Λ,a_Λ[× W_Λ onto the open set ψ(]-a_Λ,a_Λ[× W_Λ) in ℒ_d+c and such that
ψ(]-a_Λ,a_Λ[× W_Λ)∩ S=W_Λ. Since K can be covered  by finitely many W_Λ, 	we can find an open set W in S containing K and α>0 such that 

    ψ(]-2α,2α[× W)∩ S=W.

We can suppose that  W is relatively compact w.l.o.g.. Since ψ is one to one on the compact set {0}×W, reducing α, we can suppose that ψ is one to one on ]-2α,2α[×W. It follows that ψ is a diffeomorphism from U=]-2α,2α[× W onto ψ(U).
We can choose η>0 such that B(I_d+c,η)K⊂ψ([-α/4,α/4]× W). With this choice of η, for each (h,Λ)∈ B(I_d+c,η)× K, there exists  a pair (t,Γ)∈ [-α/4,α/4]× W such that g_tΓ=ψ(t,Γ)=hΛ.
If (t',Γ')∈ [-α,α]× S is such that g_t'Γ'=hΛ=g_tΓ, then Γ'=g_t-t'Γ∈ψ(]-2α,2α[× W)∩ S=W. Hence Γ'∈ W and since ψ is one to one, t'-t=0 and Γ'=Γ.
Finally, (σ(h,Λ),π(h,Λ))=(ψ_|_U)^-1(hΛ) is a continuous function of (h,Λ)∈ B(I_d+c,η)× K.




 The second lemma is
a purely measure-theoretic result.


	 Let X and Y be locally compact second countable
	metric spaces. Let μ_X and μ_Y be two measures on X and Y
	finite on compact subsets. Suppose ψ:X→ Y is a continuous map
	such that every y in Y has at most N preimages and such that for all x in X
	there exists a compact neighborhood ω_x of x with the following
	property:
	- ψ is one to one on ω_x,
	- the image by
	ψ of the measure 1_ω_xμ_X is the measure 1_ψ
		(ω_x)μ_Y. 
	Then for all nonnegative measurable functions
	f:Y→ℝ,
	
    ∫_Xf∘ψ  dμ_X≤ N∫_Yf dμ_Y.

	



Since X is a countable union of
compact sets, making use of the monotone convergence theorem, we can reduce to X
compact. Next, since for any positive Radon measure μ on Y and  for any Borel set B⊂ Y, μ(B)=sup{μ(K):K compact ⊂ B}, we can assume that f=1_K is
the indicator function of a compact subset K of Y. With these assumptions, L=ψ^-1(K) is a
compact subset of X and therefore is included in a finite union of
ω_x, 
L⊂∪_i=1^nω_x_i.
For each i, let ω_x_i^'=ω_x_i∖∪
_j<iω_x_j, L_i=ω_x_i^'∩ L and K_i
=ψ(L_i). Since L is the disjoint union of the L_i, μ
_X(L)=∑_i=1^nμ_X(L_i). Since L_i is included in
ω_x_i, μ_X(L_i)=μ_Y(K_i). Since the L_i are
disjoint, the number of ψ-preimage of any y is at least F(y)=∑
_i=1^n1_K_i(y) therefore F≤ N. It follows that

    Nμ_Y(K)     ≥∫_KF(y)dμ_Y(y)=∑_i=1^nμ_Y(K_i)
    =∑_i=1^nμ_X(L_i)=μ_X(L).






The last lemma is an easy consequence of the previous lemma and of  the definition of the induced measure μ_S.


	 Let U be a Borel subset in ℒ_d+c
	such that for all Λ in U, g_tΛ∈ U for all t in a time
	interval I_Λ of length 1 containing 0. Then
	
    μ_S(U∩ S)≤ 4Aμ(U)

	where A is the maximum number of entrance times in S of a flow trajectory
	during a time interval of length 1 (see Lemma <ref>).


The constant 4A is certainly not the best one.


Take X=[-1,1]× S,
Y=ℒ_d+c, ψ:X→ Y the map
defined by ψ(t,Λ)=g_tΛ, the measures μ_X=dt⊗μ_S and μ_Y=μ and the function f=1_U. By definition of the
induced measure, we know that the image of the restriction of dt⊗μ_S to a small enough compact neighborhood ω of any x∈ X is the restriction to
ψ(ω) of the invariant measure μ on ℒ_d+c.
Therefore by Lemma <ref>,

    Nμ(U)≥∫_S∫_-1^11_U(g_tΛ)dt dμ_S
    (Λ)

where N is the maximum number of preimages. By assumption ∫_-1
^11_U(g_tΛ)dt≥ 1 for all Λ in U and it is easy to see
that N≤ 4A. Therefore 4Aμ(U)≥μ_S(U∩ S)× 1.







  §.§.§ An example


We want to construct a co-dimension one submanifold V in ℒ_d+c
transverse to the flow g_t together with a bounded continuous function
φ:V→ℝ such that for a set  of
θ∈ M_d,c() of positive measure,  the sequence 1/n∑_k=0^n-1φ∘ R_V
^k(Λ_θ) does not converge to 1/μ_V(V)∫
_Vφ dμ_V where R_V is the first return map in V and
μ_V is the invariant measure induced by the flow. The idea is the
following. Take V an open set in S. Then μ_V is the restriction of
μ_S to V. Suppose that the open set V can be chosen in order that
for all θ∈ M_d,c(), and all k≥ 1,

    R_V^k(Λ_θ)=R_S^k(Λ_θ) (=R^k(Λ
    _θ)).

Then if φ:S→ℝ is a non negative continuous function
not identically zero with support included in V, the sequences

    1/n∑_k=0^n-1φ∘ R_V^k(Λ_θ), 1/n∑_k=0^n-1φ∘ R_S^k(Λ_θ)

converge to the same limit which cannot be equal to both 1/μ_S
(S)∫_Sφ dμ_S and 1/μ_V(V)∫_Vφ
dμ_V=1/μ_S(V)∫_Sφ dμ_S provided that μ
_S(V)<μ_S(S). So we are reduced to constructing V.

Observe that Theorem <ref> implies that for such a V, φ=1_V
is not almost everywhere continuous on S which means that the boundary of
V in S has positive measure. Moreover, it shows that the assumption about the continuity of the function φ in Theorem <ref>, cannot be dropped.



  §.§.§ Construction of V


Consider the set 𝕋 of lattices Λ_θ such that the
coefficients of θ are all in [0,1]. It is a compact subset in
ℒ_d+c containing all the lattices Λ_θ. Denote
W_ε the open ball B(I_d+c,ε) in (d+c,ℝ).
We consider the open sets

    U_n(ε)=⋃_t∈[n,n+1]g_t(W_ε𝕋)

and for a sequence (ε_n)_n∈ℕ of positive real
numbers, we consider the open set

    U=U((ε_n)_n∈ℕ)=
    ⋃_n∈ℕ
    U_n(ε_n).

Take V=S∩ U. For all t≥ 0 and all θ, g_tΛ_θ=g_tI_d+cΛ_θ is in U, hence for all k∈ℕ, we have

    R_S^k(Λ_θ)=R_V^k(Λ_θ).

So we are reduced to showing that when the sequence (ε_n)_n is
small enough,

    μ_S(V)<μ_S(S).







By definition of U_n(ε), if Λ=g_tgΛ_θ with
t∈[n,n+1], g∈ W_ε and Λ_θ∈𝕋, then

    g_sg_tgΛ_θ∈ U_n(ε)

for all s in the interval [n-t,n+1-t]. So U satisfies the assumption of the
Lemma <ref> and therefore

    μ_S(U∩ S)≤ 4A∑_n∈ℕμ(U_n(ε_n)).

Using that g_tgΛ=g_n(g_t-ngg_-(t-n))g_t-nΛ, we see
that

    U_n(ε)       ={g_tΛ:t∈[n,n+1], Λ∈ W_ε𝕋}
       ⊂ g_nW_ε^'{g_sΛ:s∈[0,1], Λ∈𝕋}

where ε^' is such that g_sW_εg_-s⊂
W_ε^' for all s in [0,1]. Furthermore, the compact set
{g_sΛ:s∈[0,1], Λ∈𝕋} has zero measure because
it has dimension cd+1 which is <(c+d)^2-1. Therefore

    lim_ε^'→ 0μ(W_ε^'{g_sΛ:s∈[0,1], Λ∈𝕋})=0,

which implies

    lim_ε→ 0μ(U_n(ε))=0.

So there exists a sequence (ε_n)_n∈ℕ such that

    ∑_n∈ℕμ(U_n(ε_n))<1/4Aμ(S)

and for such a sequence, we have μ_S(V)<μ_S(S).



 §.§ Proof of Proposition <ref>


Let φ:S→ℝ be a bounded continuous  function.

Let ε be a positive real number, let K and δ be
associated with ε by Lemma <ref>, and α and
η associated with K by Lemma <ref>.

Preliminary observation. 
Let (a_n)_n∈ be a decreasing sequence of reals numbers in ]0,η[
tending to zero and let L_a_n=B(I_d+c,a_n)× K. Since
the intersection of all the compact sets L_a_n, n∈, is L_0={I_d+c}×
K and since the map ψ(g,Λ)=φ(π(g,Λ))-φ
(Λ) is continuous, we have

    ∩_n≥ 0ψ(L_a_n)=ψ(∩_n≥ 0L_a_n)=ψ(L_0)={0}.

Therefore, for n large enough, ψ(L_a_n)⊂]-ε
,ε[ which implies there exists β>0 such that for all
Λ∈ K and all g∈ B(I_d+c,β),

    |φ(π(g,Λ))-φ(Λ)|≤ε.


Finally, let γ>0 be such that for all s≥ 0 and all h∈
B_ℋ_≤(I_d+c,γ)B_ℋ_≤^-1
(I_d+c,γ), 

    (g_shg_-s,I_d+c)≤min(δ,β,η).


For T≥ 0, Λ∈ℒ_d+c, and E⊂ S, denote

    I(T,Λ,E)={t∈[0,T]:g_tΛ∈ E}.



For almost all θ∈ M_d,c(), we can fix h_θ∈ B_ℋ_≤(I_d+c,γ) such that the conclusion of the Birkhoff ergodic theorem holds for the flow g_t or the first return map in S and the 
lattice h_θΛ_θ. Observe that h_θ=h_θ,ε 
depends on ε.
It is understood that we shall use the Birkhoff ergodic theorem in  countably many 
situations. We fix a sequence (ε_n) going to zero and for each
ε=ε_n, we use three times Birkhoff ergodic theorem and the ergodicity of the flow: for
almost all θ,

    lim_T→∞1/T∫_0^T1_U(K,δ)(g_t
    h_θΛ_θ)dt
    =1/μ(ℒ_d+c)μ(U(K,δ))≤ε,


    lim_T→∞1/TcardI(T,h_θΛ_θ,S)=1/μ_S(S)∫_Sτ_S dμ_S=μ(ℒ_d+c)/μ_S(S),

where τ_S is the first return time in S,
and

    lim_T→∞1/*cardI(T,h_θΛ_θ,S)∑_t∈ I(T,h_θΛ_θ,S)φ
    (g_th_θΛ_θ)=1/μ_S(S)μ_S(φ).


Let T be positive and let s_1<...<s_m be the elements of I(T,Λ_θ,S∖ K),
we have

    g_th_θΛ_θ=g_t-s_i(g_s_ih_θg_-s_i)g_s_iΛ_θ∈ U(K,δ)

for all t∈[ s_i,s_i+1] and we can extract a subsequence s_n_1,...,s_n_p defined by n_1=1 and n_i+1=min{j:s_j≥ s_n_i+1}. Now by Lemma <ref>, there is an absolute constant A such that there are at most
A elements of I(T,Λ_θ,S∖ K) (⊂ I(T,Λ
_θ,S)) in an interval of length 1, hence Ap≥ m. Therefore, by
(<ref>)

    m/A≤ p≤∫_0^T+11_U(K,δ)(g_th_θΛ_θ)dt≤ 2(T+1)ε

and hence

    *cardI(T,Λ_θ,S∖ K)≤ 3ATε

for T large enough: T≥ T(Λ_θ,ε). We will also
need to bound above the number of elements in I(T,h_θΛ_θ,S∖ K). Making use of (<ref>), the same way of
reasoning leads to the same result

    I(T,h_θΛ_θ,S∖ K)≤ 3ATε

for T≥ T(Λ_θ,ε).

Heart of the proof. We want to compare

    Σ_1=1/*cardI(T,Λ_θ,S)∑_t∈
    I(T,Λ_θ,S)φ(g_tΛ_θ)

with

    Σ_2=1/*cardI(T,h_θΛ_θ,S)∑_t∈ I(T,h_θΛ_θ,S)φ(g_th_θΛ_θ)

because by (<ref>), this latter sum tends to 1/μ_S(S)∫_Sφ dμ_S when T goes to infinity. We split ∑_t∈
I(T,Λ_θ,S) into two sums ∑_t∈ I(T,Λ_θ,K)
and ∑_t∈ I(T,Λ_θ,S∖ K). Observe that for t∈
I(T,Λ_θ,K), g_th_θΛ_θ=(g_th_θg_-t)g_tΛ_θ is of the form gΛ with g∈
B(I_d+c,η) and Λ∈ K, this allows to use Lemma <ref>.
We use the notations of Lemma <ref> and for t in I(T,Λ
_θ,K), we denote t^'=σ(g_th_θg_-t,g_tΛ_θ). By (<ref>), we have

    |∑_t∈ I(T,Λ_θ,K)φ(g_tΛ_θ)-∑_t∈ I(T,Λ_θ,K)φ(π(g_th_θg_-t
    ,g_tΛ_θ))|≤εcard
    I(T,Λ_θ,K).

Now,

    π(g_th_θg_-t,g_tΛ_θ)=g_-t^'g_t
    h_θg_-tg_tΛ_θ=g_t-t^'h_θΛ_θ

hence

    |∑_t∈ I(T,Λ_θ,K)φ(g_tΛ_θ)-∑_t∈ I(T,Λ_θ,K)φ(g_t-t^'h_θΛ_θ)|≤εcard
    I(T,Λ_θ,K).

Observe that the map t∈ I(T,Λ_θ,K)→ t-t^' is one to one because t^'∈
[-α/4,α/4] and the gap between two visiting times
of K is ≥α. Observe also that t-t'∈
I(T,h_θΛ_θ,S) except possibly for the first and the last element of t∈ I(T,h_θΛ_θ,S). On the one hand, it follows that

    |∑_t∈ I(T,Λ_θ,S)φ(g_tΛ_θ)-∑_t∈ I(T,h_θΛ_θ,S)φ(g_th_θΛ_θ)|
    ≤εcardI(T,Λ_θ,K)
    
    +‖φ‖ _∞(cardI(T,Λ
    _θ,K∖ S)+*cardI(T,h_θΛ_θ,S)-*cardI(T,Λ_θ,K)+2).

On the other hand, it follows that

    *cardI(T,h_θΛ_θ,S)≥*card
    I(T,Λ_θ,K)-2

and the same way of reasoning leads to

    *cardI(T,Λ_θ,S)≥*card
    I(T,h_θΛ_θ,K) -2.

Making use of (<ref>) and (<ref>), we obtain

    -2     ≤*cardI(T,h_θΛ_θ
    ,S)-*cardI(T,Λ_θ,K)=
       *cardI(T,h_θΛ_θ,S)-*card
    I(T,h_θΛ_θ,K)
         +*cardI(T,h_θΛ_θ,K)-*card
    I(T,Λ_θ,S)
         +*cardI(T,Λ_θ,S)-*card
    I(T,Λ_θ,K)
       ≤ 3 ATε+2+3ATε=6ATε+2,

hence (using (<ref>) once again)

    |∑_t∈ I(T,Λ_θ,S)φ(g_tΛ_θ)-∑_t∈ I(T,h_θΛ_θ,S)φ(g_th_θΛ_θ)|≤εcardI(T,Λ
    _θ,K)+(9Aε T+2)‖φ‖ _∞

for T≥ T(Λ_θ,ε). We obtain

    |*cardI(T,h_θΛ_θ
    ,S)-*cardI(T,Λ_θ,S)|   ≤|*cardI(T,h_θΛ_θ,S)-*card
    I(T,Λ_θ,K)|
         +|*cardI(T,Λ_θ,S∖
    K)|
       ≤ 9ATε+2

as well. Relation (<ref>) implies that card
I(T,h_θΛ_θ,S)≥ aT for T≥ T(Λ_θ,ε) where a=1/2μ(ℒ)/μ_S(S). All
together, for T≥ T(Λ_θ,ε), we obtain

    |Σ_2-Σ_1|   ≤|Σ
    _2-cardI(T,Λ_θ,S)/*card
    I(T,h_θΛ_θ,S)Σ_1| +|*cardI(T,Λ_θ,S)-*card
    I(T,h_θΛ_θ,S)/*cardI(T,h_θΛ_θ,S)||Σ_1|
       ≤1/cardI(T,h_θΛ_θ
    ,S)|∑_t∈ I(T,h_θΛ_θ,S)φ(g_t
    h_θΛ_θ)-∑_t∈ I(T,Λ_θ,S)φ(g_tΛ
    _θ)|
         +|*cardI(T,Λ_θ
    ,S)-*cardI(T,h_θΛ_θ,S)/*cardI(T,h_θΛ_θ,S)|‖φ‖ _∞
       ≤εcardI(T,Λ_θ
    ,K)+(9ATε+2)‖φ‖ _∞/card
    I(T,h_θΛ_θ,S)+(9ATε+2)‖φ‖
    _∞/cardI(T,h_θΛ_θ,S)
       ≤ε(1+18AT‖φ‖ _∞/aT)+4φ_∞/aT

which is ≪ε when T is large enough.



 §.§ Proof of Theorem <ref>


Step 1. Let us show that the restriction of f to S is integrable
with respect to μ_S.

We use Lemma <ref> with X=]0,1[× S, Y=ℒ
_d+c, the map ψ:]0,1[× S→ℒ defined by
ψ(t,Λ)=g_tΛ, the measures μ_X=dt⊗μ_S and
μ_Y=μ, and the function f. By definition of the induced measure, we
know that the image of the restriction of dt⊗μ_S to any small enough
open subset ω is the restriction to ψ(ω) of the invariant
measure μ on ℒ_d+c. Now, by Lemma <ref>, each element of ℒ_d+c has at most
A ψ-preimages, therefore by Lemma <ref>,

    ∫_0^1∫_Sf(g_tΛ)dμ_Sdt≤ A∫_ℒ_d+cfdμ.

Since f is uniformly continuous in the ℋ_≤ direction,
there exists Δ>0 such that for all Λ, and all t∈
[0,Δ], f(g_tΛ)≥ f(Λ)-1. Therefore

    ∫_0^Δ∫_S(f(Λ)-1)dμ_Sdt≤ A∫_ℒ_d+c
    fdμ,

which implies ∫_Sf(Λ)dμ_S≤μ_S(S)+A/Δ∫_ℒ_d+cfdμ<+∞.

Step 2. It is enough to prove the theorem for non negative continuous functions
φ.

Indeed, since φ is continuous almost everywhere and since |φ|≤ f with f continuous and in L^1(μ_S), for all positive
integer p, there exist two continuous functions φ_p^- and
φ_p^+ such that

    -f≤φ_p^-≤φ≤φ_p^+≤ f

and

    ∫_Sφ dμ_S-1/p≤∫_Sφ_p^-dμ_S≤∫_Sφ_p^+dμ_S≤∫_Sφ dμ_S+1/p.

Therefore, if the convergence holds for almost every θ, we have for all the functions φ_p^- and φ_p^+, 

    ∫_Sφ_p^-dμ_S     =lim_n→∞1/n∑_k=0^n-1φ_p^-∘ R^k(Λ_θ)≤liminf_n→∞1/n∑_k=0^n-1φ∘ R^k
    (Λ_θ)
       ≤limsup_p→∞1/n∑_k=0^n-1φ∘
    R^k(Λ_θ)≤lim_n→∞1/n∑
    _k=0^n-1φ_p^+∘ R^k(Λ_θ)=∫_Sφ
    _p^+dμ_S

which implies that for almost all θ,

    lim_n→∞1/n∑_k=0^n-1φ∘ R^k
    (Λ_θ)=∫_Sφ dμ_S.

Thus, we are reduced to proving the theorem for φ continuous.
Writing φ=φ^+-φ^-, we can also suppose 0≤φ≤ f.

Step 3. We prove that lim_n→∞inf1/n∑_k=0^n-1φ∘
R^k(Λ_θ)≥1/μ_S(S)∫_Sφ dμ_S for almost all θ.

 Using Proposition <ref> with the
minimum of φ and of a constant M, we obtain for almost all θ,

    lim_n→∞inf1/n∑_k=0^n-1φ∘
    R^k(Λ_θ)≥lim_n→∞1/n∑
    _k=0^n-1min(φ,M)∘ R^k(Λ_θ)=1/μ_S
    (S)∫_Smin(φ,M) dμ_S,

hence, letting M going to infinity, we obtain

    lim_n→∞inf1/n∑_k=0^n-1φ∘
    R^k(Λ_θ)≥1/μ_S(S)∫_Sφ dμ_S.




Step 4. Main step, we  bound above the sums ∑_k=0^n-1φ∘ R^k
(Λ_θ).

Since f is in L^1(μ), there exists ε^'>0 such that
for any measurable subset B in ℒ_d+c, we have

    μ(B)≤ε^'⇒∫_Bfdμ≤ε.

This allows to strengthen Lemma <ref>:


For all ε>0, there exists a compact subset K in S and
δ>0 such that 1/μ(ℒ_d+c)∫_U(K,δ)fdμ
and 1/μ(ℒ_d+c)μ(U(K,δ)) are ≤ε.


We keep all the choices and the notations of the proof of Proposition
<ref>, and we use the Birkhoff ergodic theorem with one more function:

    lim_T→∞1/T∫_0^Tf(g_th_θΛ_θ)1_U(K,δ)(g_th_θΛ_θ)dt=1/μ(ℒ_d+c)∫_U(K,δ)fdμ≤ε

so that (<ref>), (<ref>), (<ref>) and (<ref>)
hold for almost all θ (for (<ref>) observe that 0≤φ≤ f∈ L^1(μ_S)).

Since the function f is uniformly continuous in the ℋ_≤-direction, there exists κ>0 such that f(hΛ)≥ f(Λ)-1/2 for all Λ and all h∈ B_ℋ_≤(I_d+c
,κ). By choosing γ small enough we can suppose that
(g_shg_-s)∈ B_ℋ_≤(I_d+c,κ) for all s≥ 0
and all h∈ B_ℋ_≤(I_d+c,γ). Furthermore, there
exists a positive constant Δ=Δ(κ) such that g_t∈
B_ℋ_≥ 0(I_d+c,κ) for all t∈[0,Δ]. Therefore,
for all lattices Λ, all non negative real number s, all h∈
B_ℋ_≤(I_d+c,γ) and all t∈ [0,Δ], we have

    f(g_t(g_shg_-s)g_sΛ)≥ f((g_shg_-s)g_sΛ)-1/2≥ f(g_sΛ)-1

and hence

    f(g_t+shΛ)=f(g_t(g_shg_-s)g_sΛ)≥ f(g_sΛ)-1.



As in the proof of Proposition <ref>, let s_1<...<s_m be
the elements of I(T,Λ_θ,S∖ K). On the one hand g_th_θΛ_θ∈ U(K,δ) for all t∈[ s_i,s_i+1], and on the other hand, for almost
all θ,  (<ref>) and (<ref>) hold for T≥
T(Λ_θ,ε). We can suppose Δ<1 and since there
are at most A elements of I(T,Λ_θ,S∖ K) (⊂
I(T,Λ_θ,S)) in an interval of length 1, by
(<ref>) we obtain

    ∑_s∈ I(T,Λ_θ,S∖ K)φ(g_sΛ_θ)
       ≤∑_s∈ I(T,Λ_θ,S∖ K)f(g_sΛ_θ)
       ≤∑_i=1^m1/Δ∫_s_i^s_i+Δ
    (1+1_U(K,,δ)(g_th_θΛ_θ)f(g_th_θΛ_θ))dt
       ≤ m+A/Δ∫_0^T+11_U(K,δ)(g_th_θΛ_θ)f(g_th_θΛ_θ)dt

and with (<ref>) and (<ref>), this gives

    ∑_s∈ I(T,Λ_θ,S∖ K)f(g_sΛ_θ
    )≤ 3ATε+A/Δ3Tε≤ 6A/ΔTε

for all T≥ T(Λ_θ,ε).

We want to bound above

    Σ_1(T)=1/*cardI(T,Λ_θ,S)∑_t∈
    I(T,Λ_θ,S)φ(g_tΛ_θ)

with

    Σ_2(T)=1/*cardI(T,h_θΛ_θ,S)∑_t∈ I(T,h_θΛ_θ,S)φ(g_th_θΛ_θ)

because this latter sum tends to 1/μ_S(S)∫_Sφ dμ_S
when T goes to infinity. We split ∑_t∈ I(T,Λ_θ,S) into
two sums ∑_t∈ I(T,Λ_θ,K) and ∑_t∈ I(T,Λ
_θ,S∖ K). As in the proof of Proposition <ref>, for T large enough, we
have

    | I(T,h_θΛ_θ,S)-I(T,Λ_θ,S)|≤ 9ATε+2,


    I(T,h_θΛ_θ,S)≥ aT,

and

    |∑_t∈ I(T,Λ_θ,K)φ(g_tΛ_θ)-∑_t∈ I(T,Λ_θ,K)φ(g_t-t^'h_θΛ_θ)|≤εcard
    I(T,Λ_θ,K)

where t^'=σ(g_th_θg_-t,g_tΛ_θ) is
defined in Lemma <ref>. Taking into account  the first element t_min and  the last element in I(T,Λ_δ,K), the latter inequality implies that

    ∑_t∈ I(T,Λ_θ,K)φ(g_tΛ_θ)     ≤φ(g_t_minΛ_θ)+
    ∑_t∈ I(T,Λ_θ,K)∖{t_min}φ(g_t-t^'h_θΛ_θ)+εcardI(T,Λ_θ,K)
       ≤φ(RΛ_θ)+∑_t∈ I(T+1,h_θΛ_θ,S)φ(g_th_θΛ_θ)+εcardI(T,Λ_θ,K).

All together, we obtain (recall that φ≥ 0)

    Σ_1(T)     ≤1/cardI(T,Λ_θ,S)φ(RΛ_θ)+
    cardI(T+1,h_θΛ_θ
    ,S)/cardI(T,Λ_θ,S)Σ_2(T+1)+ε
       +1/*cardI(T,Λ_θ,S)∑_s∈
    I(T,Λ_θ,S∖ K)f(g_sΛ_θ)
       ≤1/cardI(T,Λ_θ,S)φ(RΛ_θ)+
    (  1+|*cardI(T+1,h_θΛ_θ,S)-*cardI(T,Λ_θ,S)/*cardI(T,Λ_θ,S)|)  Σ
    _2(T+1)+ε
       +1/*cardI(T,Λ_θ,S)6A/ΔTε
       ≤Σ_2(T+1)+1/aT-9ATε-2φ(RΛ_θ)+9ATε+2+A/aT-9ATε-2Σ_2(T+1)+(1+6A/(aT-9ATε-2)Δ)ε

and we are done. □



 §.§ Proofs of Lemma <ref>


We need an auxiliary lemma.


Let E(λ,η) be the set of lattices Λ in ℒ_d,c such
that there exist two nonzero vectors X≠± X^' of Λ in the
open ball oB_d,c(0,λ)with 1/1+η
<| X| _±/| X^'| _±<1+η or a nonzero vector X in the open ball
oB_d,c(0,λ) with | X| _±<η.
For all λ>0, we have lim_η→ 0μ(E(λ,η))=0.



Since lim_ρ→ 0μ({λ_1(Λ)≤ρ})=0, it is
enough to show that for all ρ>0, μ(E(λ,η)∩{λ
_1(Λ)≥ρ})→ 0 when η goes to 0.
Choose a Siegel reduction domain 𝒮⊂(d+c,ℝ) (see <cit.>). There is a
constant c=c(𝒮)>0 such that for all matrices M in 𝒮
and all vectors Y in ℝ^d+c, we have

    ‖ MY‖ _d,c≥ cλ_1(Λ
    )‖ Y‖ _d,c.

where Λ=Mℤ^d+c (this inequality holds for all norms with a
constant c depending only on the norm, just use the norm equivalence). It
follows that we can find a finite subset F_ρ of ℤ^d+c such
that for all matrices M in 𝒮 with λ_1(Mℤ
^d+c)≥ρ, the only Y in ℤ^d+c such that ‖
MY‖ _d,c≤λ, are in F_ρ. Therefore,
if a matrix M in 𝒮 is such that Λ=Mℤ^d+c
belongs to E(λ,η)∩{λ_1(Λ)≥ρ}, then there
exist a nonzero Y in F_ρ or two nonzero vectors Y≠± Y^' in
F_ρ with

    | MY| _±≤η

or

    1/1+η<| MY| _±/|
    MY^'| _±<1+η.

For a fixed Y or a fixed pair Y≠± Y^' of nonzero vectors in
F_ρ, the measure of the set of M in 𝒮 for which the above
inequalities hold, goes to 0 as η goes to 0. Since F_ρ is
finite and since a Siegel domain contains a fundamental domain, we are done.



Consider the set V(λ,η
,ρ)=E(λ,η)∪{λ_1(Λ)<ρ}.

Step 1:  The complementary V^C of V(λ,η,ρ) is a
closed subset of ℒ_d,c.
Let (Λ_n)_n∈ℕ
be a sequence of points of V^C which converges to Γ in ℒ_d,c. Firstly, since λ_1 is continuous, λ_1(Γ)=limλ_1(Λ_n)≥ρ. Secondly,
there is a sequence of matrices (M_n)_n∈ℕ such that Λ
_n=M_nℤ^d+c for all n∈ℕ and such that
(M_n)_n∈ℕ converges to M with Γ=Mℤ^d+c.
We have to show
that Γ is not in E(λ,η).
Let X=MY and X^'=MY^' be two nonzero vectors in Γ
with X≠± X^' and ‖ X‖ _d,c, ‖ X^'‖ _d,c<λ. When n
is large enough, X_n=M_nY and X_n^'=M_nY^' are in
the open ball oB_d,c(0,λ) and since Λ_n=M_nℤ^d+c is
not in E(λ,η), we have both

    | M_nY| _±≥η  and  | M_nY| _±/| M_nY^'| _±∉]1/1+η,1+η[.

Passing through the limit we obtain

    | X| _±≥η  and  | X| _±/| X^'| _±=| MY| _±/| MY^'|
    _±∉]1/1+η,1+η[.

Therefore Γ is not in E(λ,η).

Step 2: F=S∖ V(λ,η,ρ) is a compact subset of
S when λ≥ 2max{λ_1(Λ):Λ∈ℒ_d,c}.
Thanks to the Malher compactness theorem, it is enough to prove that
F is a closed subset of ℒ_d,c. Let (Λ_n)_n∈ℕ
be a sequence of points in F which converges to Γ in ℒ_d,c. We
want to prove that Γ is in F. By the first step, it is enough to prove
that Γ is in S. Choose a Siegel domain 𝒮. There is a
sequence of matrices M_n∈𝒮 such that Λ_n
=M_nℤ^d+c for all n∈ℕ and such that (M_n
)_n∈ℕ converges to M∈𝒮. For each n, there are
two vectors Y_1,n and Y_2,n in ℤ^d+c such that
X_1,n=M_nY_1,n and X_2,n=M_nY_2,n are the two vectors
associated with Λ_n by the definition of S. Since the matrices
M_n are all in the Siegel domain 𝒮 and since ‖
M_nY_i,n‖ _d,c=λ_1(Λ_n),
i=1,2, the sequences (Y_i,n)_n∈ℕ, i=1,2, are bounded
sequences in ℤ^d+c. Therefore, extracting subsequences, we can
suppose that the two sequences (Y_i,n)_n∈ℕ are constant:
Y_i,n=Y_i for all n, i=1,2. It follows that ‖
MY_i‖ _d,c=lim_n→∞‖
M_nY_i‖ _d,c=lim_n→∞λ
_1(Λ_n)=λ_1(Γ). Moreover

    | MY_1| _+=lim_n→∞|
    M_nY_1| _+=lim_n→∞λ_1(Λ
    _n)=λ_1(Γ)

and

    | MY_2| _-=lim_n→∞|
    M_nY_2| _-=lim_n→∞λ_1(Λ
    _n)=λ_1(Γ).

Suppose now that λ≥ 2max{λ_1(Λ):Λ∈ℒ_d,c}, then making use of the first step, we conclude that Γ is
in S.

Step 3. For a neighborhood W of I_d+c in (d+c,ℝ),
let

    U_W={g_thΛ:t∈[0,1], h∈ W, Λ∈ V(λ
    ,η,ρ)}.

Let us show that we can choose W in order that U_W⊂ V(2e^d+cλ,5e^d+cη,2e^d+cρ). It will finish the proof of Lemma <ref>.
Indeed, we first fix λ≥ 2max{λ_1(Λ):Λ∈ℒ_d,c}, next we take η and ρ such that μ
(E(2e^d+cλ,5e^d+cη))≤ε/2 and μ
({λ_1<2e^d+cρ})≤ε/2, then we take W
such that U_W⊂ V(2e^d+cλ,5e^d+cη,2e^d+cρ) and
δ such that B(I_d+c,δ)⊂ W. Now by the second step
K=S∖ V(λ,η,ρ) is compact and since U(K,δ)⊂
U_W⊂ V(2e^d+cλ,5e^d+cη,2e^d+cρ), we have
μ(U(K,δ))≤ε.

Let Λ be in V(λ,η,ρ), h in W and t∈[0,1].
We explain how to successively reduce W in order to obtain the above inclusion.

Case 1. Suppose λ_1(Λ)<ρ. We can choose W small
enough in order that ‖ hX‖ _d,c≤ 2‖ X‖ _d,c for all h in W and all
X in ℝ^d+c. This implies that λ_1(g_thΛ
)≤ 2e^d+cλ_1(Λ)<2e^d+cρ, hence g_thΛ∈
V(2e^d+cλ,5e^d+cη,2e^d+cρ).

Case 2. Suppose there exists a nonzero vector X in Λ∩oB_d,c(0,λ) with | X| _-<η (the case |
X| _+<η is similar). Call p_± the projections on the
subspaces E_± and ‖ u‖ the norm of the linear
operator u associated with the norm ‖ .‖ _d,c. The vector g_thX is in the open ball oB_d,c(0,2λ e^d+c) and
we have

    p_-g_thX=g_tp_-hp_-X+g_tp_-hp_+X,

hence

    | g_thX| _-<e^d+c‖ p_-h‖η+e^d+c‖ p_-hp_+‖λ.

We can choose W in order that ‖ p_-hp_+‖ <η/λ and ‖ p_-h‖≤ 2. Then we have | g_thX| _-
<3e^d+cη which implies that g_thΛ∈ V(2e^d+cλ
,5e^d+cη,2e^d+cρ).

Case 3. Suppose there exist two distinct nonzero vectors X and
X^' in Λ∩oB_d,c(0,λ) such that

    |  X|  _-,|  X^'|  _-≥η  and  1/1+η<|  X|  _-/|  X^'|  _-<1+η.

The case with |  .|  _+ is similar. As above,

    |  hX|  _-     <  p_-h|  X|
    _-+  p_-hp_+λ
       ≤(  p_-h  + p_-hp_+λ/η)|  X|  _-.

We can choose W in order that p_-h  +
p_-hp_+λ/η≤ 1+ η. We  also have

    |  hX^'|  _-≥‖ p_-hp_-X^'‖_d,c
    -  p_-hp_+  X^'_d,c  .

We can choose W in order that p_-hp_+λ/η≤η and 
‖ p_--p_-hp_-‖≤η. With this choice, we have

    |  hX^'|  _-≥|  X^'|  _-
    -η|  X^'|  _--η|  X^'|  _-.

It follows that

    |  g_thX|  _-/|  g_thX^'|  _-=|  hX|  _-/|  hX^'|  _-≤|  X|  _-/|  X^'|  _-×1+η/1-2η≤(1+η)^2/1-2η≤ 1+5η

when η is small enough. Inverting the role of X and X^', we get
the inequality |  hX^'|  _-/|  hX|
_-≤ 1+5η and we are done.








 §.§ Proofs of Theorems <ref> and <ref>.1.


We begin by the proof of Theorem <ref> which is more difficult.
We want to prove that for almost all θ in _d,c(ℝ),

    lim_n→∞1/nln q_n(θ)=L_d,c=1/μ
    _S(S)∫_Sρ(Λ) dμ_S(Λ)

and that

    lim_n→∞-1/nln r_n(θ)=c/dL_d,c.

Let us prove that the convergence almost everywhere of 1/nln q_n(θ) to L_d,c=1/μ_S(S)∫_Sρ
(Λ) dμ_S(Λ), implies the convergence almost everywhere of
-1/nln r_n(θ) to c/dL_d,c. By Dirichlet's theorem we know that for all θ and all integers n, q_n(θ)^cr_n+1(θ)^d≤ C where C is some universal constant (depending on the norms). Next, by the divergence part of the Khintchin-Groshev theorem (see <cit.>), we know that for almost all θ, there exist only finitely many Q∈ℤ^c such that 

    d(θ Q, ℤ^d)≤ψ(θ_ℝ^c)

where ψ(q)=1(q^cln q)^1/d. Therefore, for almost all θ, 1ln q_n(θ)≤ q_n(θ)^cr_n(θ)^d≤ C for all n large enough. It follows that for almost all θ, 

    c/nln q_n(θ)+lnln q_n(θ)/n≥ -d/nln r_n(θ)^d≥c/nln q_n(θ)-ln C/n

for all large enough n. Since the left hand side of the first inequality and the right hand side of the second inequality both converge to cL_d,c when lim_n→∞1/nln q_n(θ)=L_d,c, the sequence (-d/nln r_n(θ)^d)_n converges to the same limit for almost all θ. 
Therefore, the proof
of Theorem <ref> reduces to prove the first almost everywhere limit. 



Let us now prove the first almost everywhere limit. We need two lemmas. The
first one is clear.


For all compact sets K in ℝ^d+c and all ε>0, there
exists α>0 such that for all g∈ B(I_d+c,α) and all x∈ K,
(gx,x)≤ε.



Let Λ be in S^'∖𝒩. Then the return map R=R_S is defined on some neighborhood of Λ and is continuous at Λ.



 Consider the minimal
vectors X_0=X_0(Λ) and X_1=X_1(Λ). By definition of
S^' the only nonzero vectors B_d,c(0,λ_1(Λ))=𝒞
(X_0) are ± X_0. Therefore, there exists ε>0 such that all
X in Λ∖{0,± X_0} are at a distance ≥ε
from 𝒞(X_0). Since Λ is not in 𝒩, ±
X_0 and ± X_1 are the only nonzero vectors of Λ in the
cylinder 𝒞(X_0,X_1). Therefore reducing ε if
necessary, all X in Λ∖{0,± X_0,± X_1} are at a
distance ≥ε from 𝒞(X_0,X_1). By the above lemma
we can choose δ>0 such that
∀ g∈ B(I_d+c,δ), ∀ X∈𝒞(X_0,X_1)+B_d,c(0,1),

    max((g^-1X,X),(gX,X))≤ε/3.

It follows that for all g∈ B(I_d+c,δ), ± gX_0 are the only
nonzero vectors of gΛ in 𝒞(gX_0) and that ± gX_0
and ± gX_1 are the only nonzero vectors of Λ in 𝒞
(gX_0,gX_1). It follows that if the lattice Γ=gΛ is in the set of lattices B(I_d+c
,δ)Λ∩ S^' then X_0(Γ)=gX_0 and X_1
(Γ)=gX_1. By definition of S, the return times  from Λ and Γ are well defined and we have

    τ(Λ)   =1/d+1ln|  X_1|  _-/|
    X_0|  _+  and τ(Γ)    =1/d+1ln|  gX_1|  _-/|
    gX_0|  _+,

hence R(Λ) and R(Γ) are defined and

    |  τ(Λ)-τ(Γ)|        =1/d+1|  ln|  X_1|  _-/|  X_0|  _+|
    gX_0|  _+/|  gX_1|  _-| 
       ≤1/d+1(|  ln|  X_1|  _-/|
    gX_1|  _-|  +|  ln|  gX_0|  _+/|  X_0|  _+|  ).

which goes to zero when δ goes to zero.




We use Theorem <ref> with S^' and the function
φ:S^'→ℝ_≥ 0 defined by φ
(Λ)=ρ∘ R(Λ)=lnq_1(Λ)/q_0(Λ) when R(Λ) is defined and by φ(Λ)=0 otherwise.
Since ρ is continuous on S, the above lemma implies that φ is almost everywhere continuous on
S^'. We want to find a uniformly continuous function f:ℒ→ℝ such that |φ|≤ f.
Observe that φ is non negative. By Lemma <ref>, for
all lattice Λ∈ℒ,

    q_1(Λ)^cr_0(Λ)^d≤ C=C_d,c

where C_d,c depends only on c and d. It follows that for all Λ in
S^', we have

    φ(Λ)      =lnq_1(Λ)/q_0(Λ)
         =lnq_1(Λ)r_0(Λ)^d/c/q_0(Λ)r_0
    (Λ)^d/c
       ≤ln C^d/c-ln q_0(Λ)r_0(Λ)^d/c.

Now for Λ  in S^', we have q_0(Λ)=r_0(Λ)=λ_1(Λ),
thus

    φ(Λ)≤ln C^d/c-d+c/clnλ_1(Λ).

It is well known (see <cit.>) that the function lnλ_1 is uniformly continuous
and it is integrable on ℒ_c+d by Lemma <ref>, consequently we can use Theorem
<ref> with S^' and φ. It follows that for
almost all θ in _d,c(ℝ), we have

    lim_n→∞1/n∑_k=0^n-1φ∘ R_S^'^k(Λ_θ)=1/μ_S^'(S^')∫_S^'φ dμ_S^'

where R_S^' is the first return map to S^'. Now by Lemmas <ref> and <ref>, for almost
all θ, there is an integer k_0 such that

    φ∘ R_S^'^k(Λ_θ)=lnq_k+k_0+1(θ)/q_k+k_0(θ
    )

for all large enough k, where k_0 depends only on θ. It follows that for almost all θ 

    lim_n→∞1/n∑_k=0^n-1φ∘ R_S^'^k(Λ_θ)=
    lim_n→∞1/n∑_k=0^n-1lnq_k+k_0+1(θ)/q_k+k_0(θ
    )=
    lim_n→∞1/nln q_n(θ)

So that, the only thing left is the equality

    ∫_Sρ dμ_S=∫_S^'φ dμ_S^'.

Now, the image of μ_S^' by R is μ_S, hence

    ∫_S^'φ dμ_S^'   =∫_S^'ρ∘
    R dμ_S^'
     =∫_Sρ dμ_S.






Consider the map F:S→ defined by 

    F(Λ)=q_1^c(Λ)r_0^d(Λ)=|v_1^S(Λ)|_-^c|v_0^S(Λ)|^d_+

and call ν=ν_d,c the image of the measure 1/μ_S(S)μ_S by F.
Let φ:→ be a continuous and bounded function. We want to prove that 

    lim_n→∞1/n∑_k=0^n-1φ(β_k(θ))
    =∫_ℝφ(x)  dν(x)
 
for almost all θ∈_d,c(ℝ). 

Now by Lemmas  <ref> and <ref>, for almost all θ, 

    F(R^k(Λ_θ))=q_k+k_0+1^c(θ)r_k+k_0^d(θ)=β_k+k_0(θ)

for all k large enough. Now the function φ∘ F is bounded and continuous, thus by Theorem <ref> (or Proposition <ref>) we have for almost all θ,

    lim_n→∞1/n∑_k=0^n-1φ∘ F(R^k(Λ_θ))
       =1/μ_S(S)∫_Sφ∘ F(Λ)  dμ_S(Λ)
       =∫_ℝφ(x)   dν(x)

which  implies that

    lim_n→∞1/n∑_k=0^n-1φ(β_k(θ))
    =lim_n→∞1/n∑_k=0^n-1φ∘ F(R^k(Λ_θ))
    =∫_ℝφ(x)  dν(x)

and finishes the proof of Theorem <ref>.1.
The proof of Theorem <ref>.2. is postponed  at the end of Section 9. 




§ ON LIMINF Q_N+K^CR_N^D 


For each θ in _d,c(ℝ), we consider the sequence of best
approximation denominators (Q_n(θ))_n∈ℕ,  their norms
q_n=‖ Q_n(θ)‖ _ℝ^c, and the sequence
(r_n)_n≥ 0 defined by

    r_n=d_ℝ^d(θ Q_n,ℤ^d).

For a nonnegative integer k, call Bad_k the subset of _d,c(ℝ)
defined by

    _k(d,c)=_k={θ∈_d,c(ℝ)∖_d,c(ℚ):inf_n∈ q_n+k^c
    r_n^d>0}

(if r_n=0 for some integer n, θ is not in _k). The sequence of sets (_k)_k≥ 0 is clearly nondecreasing. The set
_0 is the usual set of badly approximable matrices, i.e., the set of matrices θ∈ M_d,c(θ) such that 

    inf{θ Q-P_^d/Q_^c:Q∈^c∖{0},P∈^d}>0.

 When d=c=1, the
classical inequality q_n+1r_n≥1/2 shows that _1
=ℝ∖ℚ while in <cit.> it is shown
that for c=1 and d≥ 2, _1 is negligible. Our first goal is to
show that _1∖_0 is nonempty for c=1 and d=2. Next we
will prove that the set

    ℬ(d,c)=ℬ=∪_k≥ 0_k

is negligible and does not depend on the choice of the norm.


If c=1 and d=2 then _1∖_0 contains uncountably many elements.


The set ℤθ+ℤ^2 is everywhere dense in
ℝ^2 for all in θ∈_1. Indeed it is known that the
first minimum of the lattice

    Λ_n=ℤ^2+ℤp_n/q_n

is ≍ r_n-1 where p_n is an integer vector such that r_n
=(q_nθ,ℤ^2). This implies that the second minimum of this
lattice λ_2(Λ_n) is ≍1/q_nr_n-1. Now, if q_nr_n-1^2≥α>0 then 1/q_nr_n-1≤r_n-1/α which goes to zero when n→∞. Therefore, λ_2(Λ_n) goes to zero which  implies that
ℤθ+ℤ^2 is everywhere dense in ℝ^2 (see
<cit.> or <cit.>).



We assume that ℝ^2 is equipped with the standard Euclidean norm .=._ℝ^2.
Set θ_0=(0,0) and θ_1=(1/5,1/5). We
construct inductively a sequence (θ_n)_n≥ 0 of rational vectors
in ℝ^2. For each n in ℕ, let Λ_n
=ℤ^2+θ_nℤ be the lattice associated with
θ_n. Observe that the least common denominator Q_n of the coordinates of the 
rational vector θ_n is the inverse of the volume of the lattice
Λ_n, Λ_n=1/Q_n (even for n=0). For 1≤ i≤ n, set

    M_i,n     =min{(qθ_n,ℤ^2)-(Q_i-1θ
    _n,ℤ^2):Q_i-1<q<Q_i},
    
    m_i,n     =(Q_i-1θ_n,ℤ^2)-(Q_iθ_n
    ,ℤ^2)

The sequence (θ_n)_n≥ 0 is constructed such that the following
properties hold for all n≥ 1:


  * Q_0=1<Q_1=5<Q_2<…<Q_n are the best approximations (denominators) of θ_n,

  * Q_n>2nQ_n-1 and given θ_0,θ_1, …,θ_n-1,
there are at least two possible choices of θ_n leading to two
different values of Q_n (to ensure that we construct an uncountable set),

  * for all 1≤ i≤ n-1, M_i,n>0 (we need to avoid the situation
where (qθ_n,ℤ^2)=(Q_i-1θ_n,ℤ^2) for
some q between Q_i-1 and Q_i),

  * ‖θ_n-θ_n-1‖≤1/8Q_n-1min{M_i,j:1≤ i<j≤ n-1},

  * ‖θ_n-θ_n-1‖≤1/8Q_n-1min{m_i,j:1≤ i≤ j≤ n-1},

  * ε_n-1=Q_n-1(θ_n-θ_n-1) is a shortest
vector of Λ_n, i.e. λ_1(Λ_n)=‖ε_n-1‖, and (-1)^n-1ε_n-1 has positive
coordinates,


  * 2λ_1(Λ_n)≤λ_2(Λ_n)≤ 30λ_1(Λ_n).


Observe that, with our choices of θ_0 and θ_1 all these conditions hold for n=1 (the conditions 3 and 4 are empty for n=1).



First, let us show that the above conditions imply that the sequence
(θ_n)_n∈ℕ converges to θ in _1∖_0. By 2 and 4, the sequence ‖θ_n-1-θ
_n‖ converges to 0 at least at a geometric rate, hence the sequence
(θ_n)_n≥ 1 converge to θ∈ℝ^2. Furthermore, by
4, for all n≥ 2,

    ‖θ-θ_n‖   ≤∑_p≥ n+1‖θ_p-θ_p-1‖
       ≤∑_p≥ n+11/8Q_p-1min{M_i,j:1≤ i<j≤ p-1}
       ≤1/4Q_nmin{M_i,n:1≤ i<n}.

Using 5 instead of 4, we obtain

    ‖θ-θ_n‖≤1/4Q_nmin{m_i,n:1≤ i≤ n}

as well. It follows that for all 1≤ i≤ n-1 and all Q_i-1<q<Q_i,
we have

    (qθ,ℤ^2)     ≥(qθ_n,ℤ^2)-q‖θ-θ_n‖
       ≥(Q_i-1θ_n,ℤ^2)+M_i,n-q‖θ
    -θ_n‖
       ≥(Q_i-1θ,ℤ^2)-Q_i-1‖θ-θ
    _n‖ +M_i,n-q‖θ-θ_n‖
       ≥(Q_i-1θ,ℤ^2)+M_i,n-2Q_i‖θ
    -θ_n‖.

Since ‖θ-θ_n‖≤1/4Q_nM_i,n,
(qθ,ℤ^2)>(Q_i-1θ,ℤ^2).
For all 1≤ i≤ n-1, we also have

    (Q_iθ,ℤ^2)     ≤(Q_iθ_n,ℤ^2
    )+Q_i‖θ-θ_n‖
       ≤(Q_i-1θ_n,ℤ^2)-m_i,n+Q_i/4Q_nm_i,n
       ≤(Q_i-1θ,ℤ^2)+Q_i-1‖θ-θ
    _n‖ -m_i,n+Q_i/4Q_nm_i,n
       ≤(Q_i-1θ,ℤ^2)+Q_i-1/4Q_nm_i,n
    -m_i,n+Q_i/4Q_nm_i,n
         <(Q_i-1θ,ℤ^2).

It follows that Q_0,Q_1,...,Q_n-1 are the first n best
approximations of θ. Therefore, (Q_n)_n≥ 0 is the sequence of
best approximations of θ and  θ_n is the rational approximation of θ associated with Q_n. The standard inequality (see for instance
<cit.>)


    λ_1(Λ_n)≍(Q_n-1θ,ℤ^2)

together with 7 imply that θ∈_1∖_0.


Let n be integer ≥ 1. Let us  explain the construction θ_n+1 given that θ
_0,...,θ_n are already constructed. First, choose a primitive point
α_n=k_nθ_n+(a_n,b_n) of Λ_n with 0≤
k_n<Q_n and (a_n,b_n)∈^2, in either ℝ_>0^2 when n is even or in
ℝ_<0^2 when n is odd. Just take α_n
a point of Λ_n in a square [x,x+1[×]0,1] with
minimal ordinate when n is even and a point of Λ_n in
a square [x,x+1[×[-1,0[ with
maximal ordinate when n is odd. Observe that ‖α_n‖ can be made arbitrarily large by choosing |x| arbitrarily large. 


Call L_n=‖α
_n‖ the length of the segment [0,α_n]. The (Euclidean) distance
between two consecutive lines of the set ℋ_n=Λ
_n+ℝα_n is

    d_n=Λ_n/L_n=1/Q_nL_n.

We can choose α_n such that

    L_n^2≥ nΛ_n,

hence L_n/d_n=L_n^2/Λ_n≥ n. There
are at least two integers p_n≥ 2 such that

    10L_n/d_n≤ p_n≤ 20L_n/d_n.

Suppose p_n is one of these and set

    ε_n=1/p_n-k_n/Q_nα_n,


    θ_n+1=θ_n+ε_n/Q_n,

and

    Q_n+1=Q_np_n-k_n.

Since by 1, Q_0=1<Q_1=5<...<Q_j are the best approximations of
θ_j, j=1,...,n, the real number min{m_i,j:1≤ i≤ j≤
n} is strictly positive. Moreover,

    ‖ε_n‖≤L_n/p_n-1≤L_n/p_n/2≤ 2L_n/10L_n/d_n≤d_n/5

and α_n can be chosen in order that d_n is arbitrarily small,
hence we can choose α_n such that ‖ε
_n‖ <‖ε_n-1‖ and

    ‖θ_n+1-θ_n‖ =1/Q_n‖ε_n‖≤1/8Q_nmin{m_i,j:1≤ i≤ j≤
    n}

which is condition 5. Next by 3, min{M_i,j:1≤ i<j≤ n} is strictly
positive. As above, it follows that α_n can be chosen such that

    ‖θ_n+1-θ_n‖ =1/Q_n‖ε_n‖≤1/8Q_nmin{M_i,j:1≤ i<j≤
    n}

which is condition 4.
Clearly Q_n+1≥ Q_n(p_n-1)≥ 2nQ_n.
Notice that the lattice Λ_n+1=ℤθ_n+1+ℤ^2
is included in ℋ_n. Next observe that

    Q_n+1θ_n+1     =(Q_np_n-k_n)(θ_n+ε_n/Q_n)
         =(Q_np_n-k_n)(θ_n+α_n/Q_n(p_n-k_n/Q_n))
         =Q_np_nθ_n-k_nθ_n+α_n
         =p_nQ_nθ_n+(a_n,b_n)∈ℤ^2.

It follows that Q_n+1Λ_n+1=l∈ℕ. On the other hand,
consider the one dimensional lattice Λ_n+1∩ℝα_n.
Because Q_nθ_n∈ℤ^2 and Q_nθ_n+1=Q_nθ_n+ε_n, it contains ε_n and is generated by a
vector v_n=ε_n/m where m is an integer. Next observe that θ_n∈Λ_n+1+α_n, hence Λ_n+1+α_n=ℋ_n. It follows that

    l/Q_n+1     =Λ_n+1=‖ v_n‖ d_n
         =‖ε_n‖/md_n=Q_nL_n/mQ_n+1d_n
         =1/mQ_n+1,

which implies m=l=1. Therefore, Λ_n+1=1/Q_n+1 and

    Λ_n+1={0,...,Q_n-1}θ_n+1+ℤε
    _n+ℤ^2.

Since ‖ε_n‖≤d_n/5, and
Λ_n+1⊂ℋ_n, ε_n is the shortest
vector of Λ_n+1. The choice of the signs for α_n now implies  that condition 6 holds. Next

    λ_1(Λ_n+1)       =‖ε_n‖ ,
    
    5‖ε_n‖   ≤ d_n≤λ_2
    (Λ_n+1)≤ d_n+‖ε_n‖ .

Since

    ‖ε_n‖≥L_n/p_n≥L_n/20L_n/d_n=d_n/20,

we obtain

    5λ_1(Λ_n+1)≤λ_2(Λ_n+1)≤ 21‖ε_n‖≤ 30λ_1(Λ_n+1)

which implies condition 7.
Let us show that Q_0,...,Q_n-1 are the first best approximations of
θ_n+1.For all 1≤ i≤ n-1 and all Q_i-1<q<Q_i,
we have

    (qθ_n+1,ℤ^2)     ≥(qθ_n,ℤ
    ^2)-q‖θ_n+1-θ_n‖
       ≥(Q_i-1θ_n,ℤ^2)+M_i,n-q‖θ
    _n+1-θ_n‖
       ≥(Q_i-1θ_n+1,ℤ^2)-Q_i-1‖θ
    _n+1-θ_n‖ +M_i,n-q‖θ_n+1-θ
    _n‖
       ≥(Q_i-1θ_n+1,ℤ^2)+M_i,n-2Q_i‖θ_n+1-θ_n‖ .

Since ‖θ_n+1-θ_n‖≤1/8Q_nM_i,n, (qθ_n+1,ℤ^2)>(Q_i-1θ_n+1
,ℤ^2) and hence M_i,n+1>0. We also have

    (Q_iθ_n+1,ℤ^2)     ≤(Q_iθ_n,ℤ
    ^2)+Q_i‖θ_n+1-θ_n‖
       ≤(Q_i-1θ_n,ℤ^2)-m_i,n+Q_i/8Q_nm_i,n
       ≤(Q_i-1θ_n+1,ℤ^2)+Q_i-1‖θ
    _n+1-θ_n‖ -m_i,n+Q_i/8Q_nm_i,n
       ≤(Q_i-1θ_n+1,ℤ^2)+Q_i-1/8Q_n
    m_i,n-m_i,n+Q_i/8Q_nm_i,n
         <(Q_i-1θ_n+1,ℤ^2).

It follows that Q_0,Q_1,...,Q_n-1 are the first n best
approximations of θ_n+1. The proof will be done once we will have
explained that Q_n and Q_n+1 are the only best approximations that
follow Q_n-1 and that M_n,n+1>0. These are the places where the sign
condition 6 plays a role. First observe that ε_n and
-ε_n are the only two shortest vectors of Λ_n+1 and
that

    ε_n=Q_nθ_n+1-Q_nθ_n

and

    -ε_n   =(Q_n+1-Q_n)θ_n+1-Q_n+1θ_n+1+Q_nθ_n
       =(Q_n+1-Q_n)θ_n+1+ a vector in ^2
    .

Together with the inequality Q_n+1-Q_n>Q_n this implies that Q_n is a best approximation
of θ_n+1 and that there is no best approximation of θ_n+1
between Q_n and Q_n+1.
Next, denoting by ≡ the equivalence modℤ^2,
we have

    Q_n-1θ_n+1     =Q_n-1(θ_n+ε_n/Q_n)=Q_n-1(θ_n-1+ε_n-1/Q_n-1+ε
    _n/Q_n)
       ≡ε_n-1+Q_n-1/Q_nε_n

and

    (Q_n-Q_n-1)θ_n+1     =(Q_n-Q_n-1)(θ_n+ε_n/Q_n)
       ≡-Q_n-1θ_n+(1-Q_n-1/Q_n)ε_n
         =-Q_n-1(θ_n-1+ε_n-1/Q_n-1)+(1-Q_n-1/Q_n)ε_n
       ≡-ε_n-1+(1-Q_n-1/Q_n)ε_n,

by the choice of the signs we obtain that

    (Q_n-1θ_n+1,ℤ^2)
    <((Q_n-Q_n-1)θ_n+1,ℤ^2).

If q is an integer ≠ Q_n-Q_n-1 lying in ]Q_n-1,Q_n[, then
qθ_n cannot be ≡±ε_n-1 which are the shortest vectors of Λ_n. Hence

    (qθ_n,ℤ^2)≥min(2‖ε_n-1‖
    ,λ_2(Λ_n))=2‖ε_n-1‖ .

It follows that

    (qθ_n+1,ℤ^2)     ≥(qθ_n,ℤ
    ^2)-q‖θ_n+1-θ_n‖
       ≥ 2‖ε_n-1‖ -q/Q_n‖ε_n‖
       ≥ 2‖ε_n-1‖ -‖ε
    _n‖ >‖ε_n-1‖
       ≥ε_n-1+Q_n-1/Q_nε_n=(Q_n-1θ_n+1,ℤ^2)

which implies both that Q_n-1 and Q_n are consecutive best
approximations of θ_n+1 and that M_n,n+1>0.





It is not clear whether the set _1 depends on the norm. However, using
an easy result about the relations between best approximation vectors
associated with two norms, we can prove:


The set ℬ(d,c) does not depend on the norms.



We give the proof only in the case c=1. When c>1 one has to extend first
the following result about best approximations:

Consider two norms N and
N^' on ℝ^d. For θ∈ℝ^d, call (q_n)_n∈  the sequence of best approximation denominators
associated with the norm N and (q_n^')_n∈  the
sequence associated with the norm N^'. Then (see <cit.>
)  there exists an integer k depending only on the norms N and
N^' such that each interval ]q_n,q_n+k], contains a best
approximation denominator q^'_m associated with the norm N^'.

It is enough to prove that ℝ^d∖_kp⊂ℝ^d∖_p^' for all p. Let θ be in
ℝ^d and n≥ k be an integer. By the above result, there
exist at least one best approximation denominator in each interval ]q_n+(j-1)k
,q_n+jk], j=0,...,p. Let q_n_j^' be the largest best
approximation denominator in each of these intervals ]q_n+(j-1)k,q_n+jk]. For each
j, we have

    r_n_j^'≤ Cr_n+jk

where C is the constant involved in the norm equivalence. Making use of the
above inequality with j=0, we obtain q_n_0+p^'r_n_0^'
d≤ C^dq_n_0+p^'r_n^d. Next q_n_0+p^'≤
q_n+kp, hence,

    q_n_0+p^'r_n_0^' d≤ C^dq_n+kpr_n^d.

It follows that liminf_n→∞q_n+kpr_n^d=0 implies
liminf_n→∞q_n+p^'r_n^' d=0.




 The set ℬ(d,c)=∪_k≥ 0_k has zero Lebesgue measure.


By the above proposition, ℬ(d,c) doesn't depend on the norms and we can suppose that ℝ^d
and ℝ^c are equipped with the standard Euclidean norms. Let us
show that for each k, _k has zero measure. We need two lemmas.


 1.
Let a<b  be two integers, let
Λ=Mℤ^d+c be a lattice in ℒ_d+c and let (Y_n)_n=a,...,b
be a sequence of vectors in ℤ^d+c. Suppose that for n=a,...,b,


  * X_n(Λ)=MY_n,

  * the only nonzero points of
Λ in the cylinder C(X_n(Λ),X_n+1(Λ)) are ±
X_n(Λ) and ± X_n+1(Λ).

Then there exists a open
neighborhood W of M in (d+c,) such that for all lattices Λ^'
=M^'ℤ^d+c∈ℒ_d+c with
M^' in W, the vectors Z_n=M'Y_n are consecutive
minimal vectors of Λ' and

    | Z_n|_+     ∈[1/2r_n(Λ),2r_n(Λ)],
    | Z_n|_-    ∈[1/2q_n(Λ),2q_n(Λ)]

for n=a,...,b.2. Suppose furthermore that a<0, b>1 and Λ∈ S.
Then for all lattices Λ^'
=M^'ℤ^d+c∈ S with M^' in W, we have X_n(Λ^')  =M^'Y_n for n=a,...,b..



1. Consider a ball B_d,c(0,R) that contains all the points MY_n,
n=a,...,b. There is a neighborhood ω of the identity matrix I_d+c such
that for all A in ω  and all X in ℝ^d+c,

    1/2‖ X‖ _d,c≤‖
    AX‖ _d,c≤ 2‖ X‖ _d,c,

so that

    X     ∉ B_d,c(0,8R)⇒ AX∉ B_d,c(0,4R)
    
    X     ∈ B_d,c(0,R)⇒ AX∈ B_d,c
    (0,2R).

Two vectors Z_n=AMY_n and Z_n+1=AMY_n+1 are consecutive
minimal vectors of AΛ as soon as

    | Z_n+1| _->| Z_n| _-
    , | Z_n+1| _+<| Z_n| _+

and the cylinder C(Z_n,Z_n+1) contains no other nonzero vector of
AΛ than ± Z_n and ± Z_n+1. Since, |
X_n+1| _->| X_n| _-, |
X_n+1| _+<| X_n| _+, by reducing
ω, we can assume | Z_n+1| _->|
Z_n| _- and | Z_n+1| _+<|
Z_n| _+, n=a,…,b. Since C(Z_n,Z_n+1)=C(AX_n,AX_n+1)⊂
B_d,c(0,2R), the image by A of a vector of Λ that is
not in the ball B_d,c(0,8R), cannot enter in the cylinder
C(AX_n,AX_n+1). Therefore, there are only finitely many X in Λ
such that AX is in C(AX_n,AX_n+1). Since by assumption all these
vectors X, except ± X_n and ± X_n+1, are at a positive distance from C(X_n,X_n+1), we obtain that
Z_n and Z_n+1 are consecutive minimal vectors by reducing once again
ω. It follows that Z_a,...,Z_b are consecutive minimal vectors of the lattice AΛ. A new reduction of ω ensures that the two  inequalities of the
lemma hold.


2. We want to see that there is no shift on the indices when Λ∈ S, a<0 and b>1. By the numbering 
convention (see Section <ref>),

    | X_0(Λ)| _+=| X_1(Λ)|
    _-, | X_-1(Λ)| _+>| X_0
    (Λ)| _-, | X_1(Λ)|
    _+<| X_2(Λ)| _-.

By a further reduction of ω, we can assume that the two inequalities
hold for the vectors Z_-1=AMY_-1, Z_0=AMY_0, Z_1=AMY_1 and Z_2=AMY_2.
Therefore, if AMℤ^d+c is in S, we must have

    X_0(AΛ)=AMY_0 and X_1(AΛ)=AMY_1

which implies that X_n(AΛ)=Z_n for n=a,....,b.




Assume that d+c≥3. Let Γ be a two dimensional lattice in
ℒ_2∖𝒩_1,1 which is in the transversal S_1,1⊂ℒ_2 and let k be a
non negative integer. Then for all positive real numbers δ, there
exists a lattice Γ_δ in
S∖𝒩 such that

    r_n(Γ_δ)     ≤2δ r_n(Γ)
    
    q_n(Γ_δ)     ≤ 2δ q_n(Γ)

for n=0,...,k.



Let Γ=Aℤ^2 be a lattice in S_1,1∖𝒩_1,1
where

    A=(
    [ a_11 a_12; a_21 a_22 ])  .

Consider the matrix M_δ∈(d+c,) defined by

    M_δ=(
    [     δ a_11          0          0          …          0     δ a_12;          0 δ^-2/d+c-2          0          …          …          0;          0          0 δ^-2/d+c-2          0          …          0;          ⋮          ⋮          ⋱          ⋱          ⋱          ⋮;          0          0          …          0 δ^-2/d+c-2          0;     δ a_21          0          …          …          0     δ a_22 ])  .

    Let (U_n=(u_n,1,u_n,2))_n∈ℤ be the sequence of vectors in ℤ^2
such that (X_n(Γ)=AU_n)_n∈ is the sequence of minimal vectors of
Γ. For each n∈ℤ, let Y_n be the element of
ℤ^d+c defined by y_1=u_n,1, y_2=...=y_d+c-1=0 and
y_d+c=u_n,2. If δ>0 is small enough, then for all
Z∈ℤ^d+c not in the e_1+ e_d+c-plane, we have

    ‖ M_δZ‖ _d,c=max(|
    M_δZ| _+,| M_δZ| _-)≥δ^-2/d+c-2≥max(2δr_-1(Γ),2δq_k+1(Γ)).

It follows that none of these vectors M_δZ are in one of the
cylinders C(M_δY_n,M_δY_n+1), n=-1,...,k.
Therefore the vectors X_n=M_δY_n, n=-1,...,k+1 are all
consecutive minimal vectors of Λ_δ=M_δℤ^d+1 and Λ_δ is in S. With the numbering
convention, we have X_n(Λ_δ)=X_n for all
n=-1,...,k+1.
Now we fix δ small enough. By the previous lemma applied to
Λ_δ, there is a sequence of matrices (M_p)_p in
S∖𝒩 which converges to M_δ and such that for all
p,

    X_n(M_pℤ^d+1)=M_pY_n

n=-1,...,k. When p goes to infinity,

    r_n(M_pℤ^d+1)       =| M_pZ_n|
    _+→| M_δZ_n| _+=δ r_n
    (Γ),
    
    q_n(M_pℤ^d+1)       =| M_pZ_n|
    _-→| M_δZ_n| _-=δ q_n(Γ)

for n=-1,...,k. So we can take Γ_δ=M_pℤ^d+1
for some p large enough.




Let k and η>0 be fixed. We want to prove that the set of θ in
_d,c(ℝ) such that

    liminf_n→∞q_n+k^c(θ)r_n^d(θ)≤η

has full measure.
By Lemma <ref>, it is enough to show that

    liminf_n→∞q_n+k^c(Λ_θ)r_n^d(Λ
    _θ)≤η

for almost all θ.
Fix a two-dimensional lattice Γ in S_1,1∖𝒩_1,1
and let δ be a positive real number with δ≤1/4
(η/q_k^c(Γ)r_0^d(Γ))^1/d+c. By 
Lemma <ref>, there exists a
lattice Λ_δ  in S∖𝒩 such that

    r_n(Λ_δ)    ≤2δ r_n(Γ)
    
    q_n(Λ_δ)    ≤ 2δ q_n(Γ)

for n=0,...,k. Hence,

    q_k^c(Λ_δ)r_0^d(Λ_δ)≤η/2^d+c.

By Lemma <ref>, there exists an open neighborhood W of
Λ_δ such that for all Λ in W and some integer m(Λ), we have both

    r_m(Λ)(Λ)≤ 4δ r_0(Γ)

and 

    q_m(Λ)+k^c(Λ)r_m(Λ)^d(Λ)≤η.



Let us show that, if for a given lattice Λ, there exists a sequence (t_n
)_n∈ℕ going to infinity such that g_t_nΛ∈ W for
all n∈ℕ, then

    liminf_n→∞q_n+k^c(Λ)r_n^d(Λ)≤η.

Indeed, if g_t_nΛ∈ W, then for some integer m(Λ,t_n), we have 

    (e^-dt_nq_m(Λ,t_n)+k(Λ))^c(e^ct_nr_m(Λ,t_n)(Λ))^d≤η.
 
So, the only thing to see is that m(Λ,t_n)→∞ when n→∞. 
Now e^dt_nr_m(Λ,t_n)(Λ)≤ 4δ r_0(Γ),
hence r_m(Λ,t_n)(Λ) goes to zero when m goes
to infinity which implies that m(Λ,t_n) goes to infinity.

Making use of the Birkhoff ergodic theorem with the flow g_t, the proof
would be already finished if our goal were
liminf_n→∞q_n+k^c(Λ)r_n^d(Λ)≤η
for almost all lattices. However we want an “almost all" with respect of the Lebesgue measure of _d,c(). 
 

Let U  be a relatively compact nonempty open set in ℒ_d+c such that
U⊂ W. 
One can find
a neighborhood V of I_d+c in ℋ_≤ such
that for all
θ∈_d,c(ℝ), all t≥ 0 and all h∈ V, we have

    g_thΛ_θ=(g_thg_-t)g_tΛ_θ∈ U⟹
    g_tΛ_θ∈ W.

Call 𝒱 the set of θ such that g_tΛ_θ∉
W for all t large enough. By the choices of U and V, for all h∈ V and all
θ∈𝒱, g_thΛ_θ∉ U for all t large
enough. If the Lebesgue measure of 𝒱 were nonzero then the set of
lattices of the form g_shΛ_θ with s∈[0,1], h∈
V and θ∈𝒱, would have a nonzero measure. Now, by the Birkhoff ergodic theorem,
for almost all lattices Λ, there exists a sequence t_n→∞ such that g_t_nΛ∈ U for all n, therefore
𝒱 has zero measure.



 By the proof of the first part of Theorem <ref>, we know that the measure ν_d,c is the image of the measure 1/μ_S(S)μ_S by the map F:S→ defined by 
	
    F(Λ)=q_1^c(Λ)r_0^d(Λ)=|v_1^S(Λ)|_-^c|v_0^S(Λ)|^d_+.

We want to prove that the support of the measure ν_d,c contains zero, i.e., that ν_d,c([0,η])>0 for all η>0. By the Birkhoff ergodic theorem and by the definition of ν_d,c, it is enough to prove that 

    lim_n→∞1/n∑_i=0^n-11_[0,η](F(R^i(Λ)))>0

for almost all Λ∈ S. 
By Lemmas <ref> and <ref>,
there exists a non empty  open set W in S such that 

    | v_1^S(Λ)|^c_-| v_0^S(Λ)|^d_+≤η

for all Λ∈ W. Hence 1_W≤ 1_[0,η]∘ F.
 By the Birkhoff ergodic theorem,
 for almost
all Λ in S

    lim_n→∞1/n∑_i=1^n1_W∘ R^i
    (Λ)=1/μ_S(S)∫_S1_W dμ_S=a>0.

therefore,

    lim_n→∞1/n∑_i=0^n-11_[0,η](F(R^i(Λ)))≥ a>0.





§ MISCELLANEOUS QUESTIONS



  * 

In Theorems <ref> and <ref>, we assume that ^d and ^c are equipped with standard Euclidean norms. Clearly, Theorems 1 and 2 are valid for any pair of Euclidean norms and by  Theorem <ref>, the Levy constant does not depend on the particular choices of the Euclidean norms. However, we do not know what happens for a pair of general norms. Are these theorems valid when ^d and ^c are equipped with any norm?
If Theorem <ref> is valid for any norm, does the Levy's constant depend on the norm?


  * Is the measure ν_d,c in Theorem <ref>, absolutely continuous with respect to Lebesgue measure?

Is the support of ν_d,c an interval? Compute the measure ν_2,1.


  * Suppose c=1. 
Consider a flow (g_t)_t∈ defined by the matrices

    g_t=diag(e^a_1t,…,e^a_dt,e^-dt) ∈ (d+1,ℝ)

where the a_is are positive real numbers with sum d.
Best approximation vectors of θ∈^d with respect to the flow (g_t)_t∈  can be defined as follows. A nonzero vector X in ^d+1 is a best approximation vector of θ if there exists t≥0 such that the interior of the ball B(0,g_tM_θX_^d+1)⊂^d× contains no nonzero vector  of the lattice g_tM_θ^d+1 (equivalently g_tM_θX_^d+1=λ_1(g_tM_θ^d+1)). Arranging the set of best approximation vector according to their heights, we obtain a sequence (X_n(θ))_n∈ of best approximation vectors associated with θ. Does Theorem <ref> hold for these new sequences of best approximation vectors?

  * For a fixed k≥ 1, does the set

    _k(d,c)=_k={θ∈_d,c(ℝ)∖_d,c(ℚ):inf_n∈ q_n+k^c(θ)
    r_n^d(θ)>0}

depend on the norm used to define best approximations vectors?

Observe that by Proposition <ref>, the union ∪_k≥ 1_k does not depend on the choice  of the norm.

  * Since  _1(d,c) contains the set of badly approximable matrices _0(d,c), it has full Hausdorff dimension. But what are the Hausdorff dimensions of  _1(d,c)∖_0(d,c) and of ∪_k≥ 0_k(d,c)∖_0(d,c)?
Has _1(d,c)∖_0(d,c)  full Hausdorff dimension? 

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