Control and its applications in additive combinatorics

TL;DR

This paper establishes new quantitative bounds on additive structures under an $L^3$ control assumption, leading to improvements in sum-product estimates, the Balog-Szemerédi-Gowers theorem, and convex set growth.

Contribution

It introduces novel bounds under an $L^3$ control condition, advancing understanding in additive combinatorics and related problems.

Findings

Improved bounds for the sum-product problem

Enhanced results for the Balog-Szemerédi-Gowers theorem

Stronger estimates on additive growth of convex sets

Abstract

We prove new quantitative bounds on the additive structure of sets obeying an $L^3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemer\'{e}di-Gowers theorem, and the additive growth of convex sets.