A Freĭman-type theorem for locally compact abelian groups

[Une généralisation du théorème de Freĭman aux groupes abéliens localement compacts]

Sanders, Tom

Annales de l'Institut Fourier, Tome 59 (2009), p. 1321-1335 / Harvested from Numdam

Résumé

Soit $G$ un groupe abélien localement compact muni d’une mesure de Haar $\mu $. La $\delta $-boule $B_{\delta }$ pour une pseudo-métrique continue et invariante par translation sera dite de dimension d si $\mu (B_{2\delta ^{\prime}}) \le 2^d\mu (B_{\delta ^{\prime}})$ pour tout $\delta ^{\prime} \subset (0,\delta ]$. Nous montrons que si $A$ est un voisinage compact symétrique de l’identité tel que $\mu (nA) \le n^d\mu (A)$ pour tout $n \ge d \log d$, 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 $\mu (B) \le \exp (O(d\log d))\mu (A)$.

Abstract

Suppose that $G$ is a locally compact abelian group with a Haar measure $\mu $. The $\delta $-ball $B_{\delta }$ of a continuous translation invariant pseudo-metric is called $d$-dimensional if $ \mu (B_{2\delta ^{\prime}}) \le 2^d\mu (B_{\delta ^{\prime}})$ for all $\delta ^{\prime} \subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu (nA) \le n^d\mu (A)$ for all $n \ge 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 $\mu (B) \le \exp (O(d\log d))\mu (A)$.

Détail

Publié le : 2009-01-01

MR 2566962 | Zbl 1179.43002

DOI : https://doi.org/10.5802/aif.2465
Classification:  43A25,  11B25
Mots clés: théorème de Freĭman, transformée de Fourier, boules dans des pseudo-métriques, croissance polynomiale

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