On sumsets of convex sets

TL;DR

This paper investigates the sumsets of convex sets of real numbers, establishing a lower bound on the size of the difference set in relation to the original set, with implications for additive combinatorics.

Contribution

It provides a new lower bound on the size of difference sets of convex sets, advancing understanding of their additive structure.

Findings

|A-A| \,\gg\, |A|^{8/5} \log^{ -2/5} |A|

Convex sets have significantly larger difference sets than previously known.

The result improves bounds in additive combinatorics for convex sets.

Abstract

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i - a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5} |A|.