Title: New results on sum-product type growth over fields,
Abstract: We prove a range of new sum-product type growth estimates over a general
field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$.
They are unified by the theme of "breaking the $3/2$ threshold", epitomising
the previous state of the art. These estimates stem from specially suited
applications of incidence bounds over $\mathbb{F}$, which apply to higher
moments of representation functions.
We establish the estimate $|R[A]| \gtrsim |A|^{8/5}$ for cardinality of the
set $R[A]$ of distinct cross-ratios defined by triples of elements of a
(sufficiently small if $\mathbb{F}$ has positive characteristic, similarly for
the rest of the estimates) set $A\subset \mathbb{F}$, pinned at infinity. The
cross-ratio naturally arises in various sum-product type questions of
projective nature and is the unifying concept underlying most of our results.
It enables one to take advantage of its symmetry properties as an onset of
growth of, for instance, products of difference sets. The geometric nature of
the cross-ratio enables us to break the version of the above threshold for the
minimum number of distinct triangle areas $Ouu'$, defined by points $u,u'$ of a
non-collinear point set $P\subset \mathbb{F}^2$.
Another instance of breaking the threshold is showing that if $A$ is
sufficiently small and has additive doubling constant $M$, then $|AA|\gtrsim
M^{-2}|A|^{14/9}$. This result has a second moment version, which allows for
new upper bounds for the number of collinear point triples in the set $A\times
A\subset \mathbb{F}^2$, the quantity often arising in applications of geometric
incidence estimates.