An update on the sum-product problem

TL;DR

This paper advances sum-product estimates over the real numbers by improving bounds and introducing new techniques, including results for convex sets and sumsets involving products, thereby contributing to additive combinatorics.

Contribution

It provides improved sum-product bounds over the reals, streamlines previous arguments, and introduces new observations, including bounds for convex sets and sumsets involving products.

Findings

+A| and |AA| are at least |A|^{4/3 + 2/1167 - o(1)}.

|AA+AA|  at least |A|^{127/80 - o(1)}.

For convex sets A, |A+A|  at least |A|^{30/19 - o(1)}.

Abstract

We improve the best known sum-product estimates over the reals. We prove that \[ \max(|A+A|,|AA|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,, \] for a finite $A\subset \mathbb R$, following a streamlining of the arguments of Solymosi, Konyagin and Shkredov. We include several new observations to our techniques. Furthermore, \[ |AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,. \] Besides, for a convex set $A$ we show that \[ |A+A|\geq |A|^{\frac{30}{19}-o(1)}\,. \] This paper is largely self-contained.