A Freĭman-type theorem for locally compact abelian groups

Sanders, Tom

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 $\mu$. La $\delta$-boule $B_{\delta }$ pour une pseudo-métrique continue et invariante par translation sera dite de dimension d si $\mu \left(B_{2{\delta }^{\prime }}\right)\le 2^d\mu \left(B_{{\delta }^{\prime }}\right)$ pour tout ${\delta }^{\prime }\subset (0,\delta ]$. Nous montrons que si $A$ est un voisinage compact symétrique de l’identité tel que $\mu \left(nA\right)\le n^d\mu \left(A\right)$ pour tout $n\ge dlogd$, alors $A$ est contenu dans une boule $B$ de dimension $O\left(d{log}^{3}d\right)$ et de rayon strictement positif pour une pseudo-métrique continue et invariante par translation  ; de plus $\mu \left(B\right)\le exp\left(O\right(dlogd\left)\right)\mu \left(A\right)$.

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 $\mu \left(B_{2{\delta }^{\prime }}\right)\le 2^d\mu \left(B_{{\delta }^{\prime }}\right)$ for all ${\delta }^{\prime }\subset (0,\delta ]$. We show that if $A$ is a compact symmetric neighborhood of the identity with $\mu \left(nA\right)\le n^d\mu \left(A\right)$ for all $n\ge dlogd$, then $A$ is contained in an $O\left(d{log}^{3}d\right)$-dimensional ball, $B$, of positive radius in some continuous translation invariant pseudo-metric and $\mu \left(B\right)\le exp\left(O\right(dlogd\left)\right)\mu \left(A\right)$.

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

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