On a theorem of Schoen and Shkredov on sumsets of convex sets

TL;DR

This paper establishes a new lower bound on the size of sumsets of convex sets of real numbers, improving understanding of their additive structure and implications for related sum-product problems.

Contribution

It proves a novel lower bound on sumsets of convex sets and explores related sumset properties and applications to sum-product problems.

Findings

Lower bound |A+A| ≫ |A|^{14/9} / (log|A|)^{2/9} for convex sets

Analysis of sumsets of different summands

Application to sum-product-type problems

Abstract

A set of reals $A=\{a_1,...,a_n\}$ labeled in increasing order is called convex if there exists a continuous strictly convex function $f$ such that $f(i)=a_i$ for every $i$. Given a convex set $A$, we prove \[|A+A|\gg\frac{|A|^{14/9}}{(\log|A|)^{2/9}}.\] Sumsets of different summands and an application to a sum-product-type problem are also studied either as remarks or as theorems.