A robust version of Freiman's $3k-4$ Theorem and applications

TL;DR

This paper extends Freiman's $3k-4$ theorem to a robust setting, providing new structural insights for sumsets with small doubling constants and applications to inequalities in additive combinatorics.

Contribution

It introduces a robust version of Freiman's $3k-4$ theorem applicable to restricted sumsets with small doubling constants, with broader applications in additive combinatorics.

Findings

Robust version of Freiman's $3k-4$ theorem established.

Structural results for sets with nearly equal sumset sizes derived.

Applications to Riesz-Sobolev inequality in discrete and continuous contexts.

Abstract

We prove a robust version of Freiman's $3k - 4$ theorem on the restricted sumset $A+_{\Gamma}B$, which applies when the doubling constant is at most $\tfrac{3+\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = -B$. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz-Sobolev inequality.