Higher Convexity and Iterated Second Moment Estimates

TL;DR

This paper establishes new bounds on the number of solutions to additive equations over sufficiently convex or near-convex sets, demonstrating that higher convexity leads to improved estimates and extending previous results with explicit dependencies.

Contribution

The paper introduces bounds for solutions to additive equations over near-convex sets, utilizing higher convexity assumptions and a novel approach based on Garaev's idea instead of Szemerédi-Trotter.

Findings

Bounds improve exponentially with the number of terms in the equation

Explicit dependencies on additive doubling parameters are provided

Higher convexity is shown to be necessary for these bounds

Abstract

We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive doubling under a convex function with sufficiently many strictly monotone derivatives. We show, roughly, that every time the number of terms in the equation is doubled, an additional saving of $1$ in the exponent of the trivial bound $N^{2k-1}$ is made, starting from the trivial case $k=1$. In the context of near-convex sets we also provide explicit dependencies on the additive doubling parameters. Higher convexity is necessary for such bounds to hold, as evinced by sets of perfect powers of consecutive integers. We exploit these stronger assumptions using an idea of Garaev, rather than the ubiquitous Szemer\'edi-Trotter theorem, which has not been adapted in earlier results to embrace higher convexity. As an application we prove small improvements for the best known bounds for sumsets of convex sets under additional convexity assumptions.