no. 4
A Freĭman-type theorem for locally compact abelian groups
[Une généralisation du théorème de Freĭman aux groupes abéliens localement compacts]
Sanders, Tom1
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.

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 2δ ′ )≤2 d μ(B δ ′ ) pour tout δ ′ ⊂(0,δ]. Nous montrons que si A est un voisinage compact symétrique de l’identité tel que μ(nA)≤n d μ(A) pour tout n≥dlogd, alors A est contenu dans une boule B de dimension O(dlog 3 d) et de rayon strictement positif pour une pseudo-métrique continue et invariante par translation  ; de plus μ(B)≤exp(O(dlogd))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 2δ ′ )≤2 d μ(B δ ′ ) for all δ ′ ⊂(0,δ]. We show that if A is a compact symmetric neighborhood of the identity with μ(nA)≤n d μ(A) for all n≥dlogd, then A is contained in an O(dlog 3 d)-dimensional ball, B, of positive radius in some continuous translation invariant pseudo-metric and μ(B)≤exp(O(dlogd))μ(A).

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

Sanders, Tom 1

1 University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WA (England) 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2465/

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