Expanding polynomials for sets with additive structure

TL;DR

This paper establishes new expansion bounds for specific bivariate polynomials over sets with small sumsets, advancing understanding in additive combinatorics and related problems.

Contribution

It provides novel expansion bounds for polynomials of the form f(x,y)=g(x+p(y))+h(y) on sets with small sumsets, using an adapted proximity approach.

Findings

Derived explicit lower bounds for polynomial expansions on small sumsets.

Applied bounds to additive combinatorics and distinct distances problems.

Extended the proximity approach to incorporate sumset sizes.

Abstract

The expansion of bivariate polynomials is well-understood for sets with a linear-sized product set. In contrast, not much is known for sets with small sumset. In this work, we provide expansion bounds for polynomials of the form $f(x, y) = g(x + p(y)) + h(y)$ for sets with small sumset. In particular, we prove that when $|A|$, $|B|$, $|A + A|$, and $|B + B|$ are not too far apart, for every $\varepsilon > 0$ we have \[|f(A, B)| = \Omega\left(\frac{|A|^{256/121 - \varepsilon}|B|^{74/121 - \varepsilon}}{|A + A|^{108/121}|B + B|^{24/121}}\right).\] We show that the above bound and its variants have a variety of applications in additive combinatorics and distinct distances problems. Our proof technique relies on the recent proximity approach of Solymosi and Zahl. In particular, we show how to incorporate the size of a sumset into this approach.