A Freĭman-type theorem for locally compact abelian groups

Sanders, Tom

doi:10.5802/aif.2465

A Freĭman-type theorem for locally compact abelian groups
[Une généralisation du théorème de Freĭman aux groupes abéliens localement compacts]

Sanders, Tom ¹

¹ University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WA (England)

Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1321-1335.

Résumé
Abstract

Soit $G$ un groupe abélien localement compact muni d’une mesure de Haar $μ$ . La $δ$ -boule $B_{δ}$ pour une pseudo-métrique continue et invariante par translation sera dite de dimension d si $μ (B_{2 δ^{'}}) \leq 2^{d} μ (B_{δ^{'}})$ pour tout $δ^{'} \subset (0, δ]$ . Nous montrons que si $A$ est un voisinage compact symétrique de l’identité tel que $μ (n A) \leq n^{d} μ (A)$ pour tout $n \geq d log d$ , alors $A$ est contenu dans une boule $B$ de dimension $O (d {log}^{3} d)$ et de rayon strictement positif pour une pseudo-métrique continue et invariante par translation ; de plus $μ (B) \leq exp (O (d log d)) μ (A)$ .

Suppose that $G$ is a locally compact abelian group with a Haar measure $μ$ . The $δ$ -ball $B_{δ}$ of a continuous translation invariant pseudo-metric is called $d$ -dimensional if $μ (B_{2 δ^{'}}) \leq 2^{d} μ (B_{δ^{'}})$ for all $δ^{'} \subset (0, δ]$ . We show that if $A$ is a compact symmetric neighborhood of the identity with $μ (n A) \leq n^{d} μ (A)$ for all $n \geq d log d$ , then $A$ is contained in an $O (d {log}^{3} d)$ -dimensional ball, $B$ , of positive radius in some continuous translation invariant pseudo-metric and $μ (B) \leq exp (O (d log d)) μ (A)$ .

MR Zbl

DOI : 10.5802/aif.2465

Classification : 43A25, 11B25
Keywords: Freĭman’s theorem, Fourier transform, balls in pseudo- metrics, polynomial growth
Mot clés : théorème de Freĭman, transformée de Fourier, boules dans des pseudo-métriques, croissance polynomiale

Affiliations des auteurs :

Sanders, Tom ¹

¹ University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WA (England)

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     title = {A {Fre\u{i}man-type} theorem for locally compact abelian groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1321--1335},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {4},
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     doi = {10.5802/aif.2465},
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Sanders, Tom. A Freĭman-type theorem for locally compact abelian groups. Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1321-1335. doi : 10.5802/aif.2465. http://www.numdam.org/articles/10.5802/aif.2465/

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