Quantitative stability for sumsets in $R^n$

TL;DR

This paper studies the stability of sumset measure equality in ^n, showing that if the measure difference between A+A and 2A is small, then A is close to its convex hull, with explicit bounds in any dimension.

Contribution

It provides an explicit quantitative stability result for sumsets in ^n, linking measure differences to geometric proximity to convex sets.

Findings

Explicit bounds on measure difference imply closeness to convex hull

Stability result holds in arbitrary dimension

Quantitative relationship between sumset measure and set convexity

Abstract

Given a measurable set $A\subset \R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(|A+A|-|2A|)/|A|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(|A+A|-|2A|)/|A|$.