Title: On subtrees of the representation tree in rational base numeration systems,
Abstract: Every rational number p/q defines a rational base numeration system in which
every integer has a unique finite representation, up to leading zeroes. This
work is a contribution to the study of the set of the representations of
integers. This prefix-closed subset of the free monoid is naturally represented
as a highly non-regular tree. Its nodes are the integers, its edges bear labels
taken in {0,1,...,p-1}, and its subtrees are all distinct.
We associate with each subtree (or with its root n) three infinite words. The
bottom word of n is the lexicographically smallest word that is the label of a
branch of the subtree. The top word of n is defined similarly. The span-word of
n is the digitwise difference between the latter and the former.
First, we show that the set of all the span-words is accepted by an infinite
automaton whose underlying graph is essentially the same as the tree itself.
Second, we study the function that computes for all n the bottom word
associated with n+1 from the one associated with n, and show that it is
realised by an infinite sequential transducer whose underlying graph is once
again essentially the same as the tree itself.
An infinite word may be interpreted as an expansion in base p/q after the
radix point, hence evaluated to a real number. If T is a subtree whose root is
n, then the evaluations of the labels of the branches of T form an interval of
$\mathbb{R}$. The length of this interval is called the span of n and is equal
to the evaluation of the span-word of n. The set of all spans is then a subset
of R and we use the preceding construction to study its topological closure. We
show that it is an interval when p is greater than or equal to 2q-1, and a
Cantor set of measure zero otherwise.