# Quantitative stability for sumsets in ${ℝ}^{n}$

• Volume: 017, Issue: 5, page 1079-1106
• ISSN: 1435-9855

top

## Abstract

top
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 $\left(|A+A|-|2A|\right)/|A|$ 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 $\left(|A+A|-|2A|\right)/|A|$.

## How to cite

top

Figalli, Alessio, and Jerison, David. "Quantitative stability for sumsets in $\mathbb {R}^n$." Journal of the European Mathematical Society 017.5 (2015): 1079-1106. <http://eudml.org/doc/277337>.

@article{Figalli2015,
abstract = {Given a measurable set $A\subset \mathbb \{R\}^n$ 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 $(|A+A|-|2A|)/|A|$.},
author = {Figalli, Alessio, Jerison, David},
journal = {Journal of the European Mathematical Society},
keywords = {quantitative stability; sumsets; Freiman’s theorem; sumsets; quantitative stability; Freiman's theorem},
language = {eng},
number = {5},
pages = {1079-1106},
publisher = {European Mathematical Society Publishing House},
title = {Quantitative stability for sumsets in $\mathbb \{R\}^n$},
url = {http://eudml.org/doc/277337},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Figalli, Alessio
AU - Jerison, David
TI - Quantitative stability for sumsets in $\mathbb {R}^n$
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 5
SP - 1079
EP - 1106
AB - Given a measurable set $A\subset \mathbb {R}^n$ 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 $(|A+A|-|2A|)/|A|$.
LA - eng
KW - quantitative stability; sumsets; Freiman’s theorem; sumsets; quantitative stability; Freiman's theorem
UR - http://eudml.org/doc/277337
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.