Ensembles de petite somme, structure de sous-criticit\'e

TL;DR

This paper generalizes previous results on the measure of sumsets of bounded real sets, providing a structural understanding near the case of equality, advancing the theory of sumset measures.

Contribution

It extends de Roton's structural results to a broader context near the equality case for sumset measure bounds.

Findings

Established a structural result near the equality case for sumset measure bounds.

Generalized previous work by Ruzsa and de Roton on sumset measures.

Provided insights into the structure of critical sets in sumset inequalities.

Abstract

If $A$ and $B$ are two bounded sets of reals, Ruzsa proved a precise lower bound of the measure of the sumset $A+B$ involving the ratio $\lambda(A)/\lambda(B)$. De Roton established a structural result about the critical sets of this lower bound. Here, we prove a generalization of de Roton's work by establishing a result in a neighborhood of the case of equality.