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

Alessio Figalli; David Jerison

Journal of the European Mathematical Society (2015)

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

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topFigalli, 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 -

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