Note on the Theorem of Balog, Szemer\'edi, and Gowers

TL;DR

This paper refines the Balog-Szemerédi-Gowers theorem by identifying the largest structured subset within an additive set with high energy, providing bounds on subset size and difference set.

Contribution

It establishes the optimal size and structure of subsets in additive sets with high energy, improving understanding of additive combinatorics.

Findings

Identifies a large structured subset within high-energy additive sets.

Provides bounds on the size of the subset relative to the original set.

Shows the difference set of the subset is controlled by a polynomial in K.

Abstract

We prove that every additive set $A$ with energy $E(A)\ge |A|^3/K$ has a subset $A'\subseteq A$ of size $|A'|\ge (1-\varepsilon)K^{-1/2}|A|$ such that $|A'-A'|\le O_\varepsilon(K^{4}|A'|)$. This is, essentially, the largest structured set one can get in the Balog-Szemer\'edi-Gowers theorem.