# New results on sum-product type growth over fields

## TL;DR

This paper advances sum-product growth estimates over fields, particularly finite fields, by leveraging incidence bounds and the cross-ratio concept to surpass the classical 3/2 growth threshold in various geometric and algebraic contexts.

## Contribution

It introduces new sum-product growth bounds over fields using incidence geometry and cross-ratio techniques, breaking the 3/2 threshold in multiple sum-product problems.

## Key findings

- Established |R[A]| rac{8}{5} |A|^{8/5} growth for cross-ratio sets.
- Proved growth in triangle areas surpassing the 3/2 threshold.
- Derived bounds on product sets related to additive doubling constants.

## 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.