Control and its applications in additive combinatorics

TL;DR

This work introduces and exploits $L^3$ control, quantified by $\kappa$, to study additive structure in finite subsets of abelian groups. The authors derive nontrivial bounds linking energy, sumsets, and difference sets under the control hypothesis, namely $E(A) \ll_\varepsilon \kappa^{27/50-\varepsilon}|A|^3$, $|A+A| \gg_\varepsilon \kappa^{-11/19+\varepsilon}|A|$, and $|A-A| \gg_\varepsilon \kappa^{-2506/4175+\varepsilon}|A|$, and show these bounds propagate to key questions like convex-set growth and the sum-product problem. They apply a unified control framework to convex sets (where $\kappa \lesssim |A|^{-1}$), sharpen sum-product exponents (yielding $\max(|A+A|,|AA|) \gg_\varepsilon |A|^{c-\varepsilon}$ with $c \approx 1.33543$), and improve Balog–Szemerédi–Gowers-type conclusions (e.g., a large subset with small doubling when $E(A)$ is large). Importantly, the paper emphasizes elementary techniques (pigeonhole principle and Hölder), highlighting a path to further quantitative improvements through the notion of control. The results unify disparate lines of the literature and illuminate the deep connections between additive structure, convexity, and incidence-geometric methods.

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-Szemerédi-Gowers theorem, and the additive growth of convex sets.

Theorems & Definitions (56)

• Definition 1: Control
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 1