A Freĭman-type theorem for locally compact abelian groups
Annales de l'Institut Fourier, Volume 59 (2009) no. 4, p. 1321-1335

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 ndlogd, 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).

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 ndlogd, 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).

DOI : https://doi.org/10.5802/aif.2465
Classification:  43A25,  11B25
Keywords: Freĭman’s theorem, Fourier transform, balls in pseudo- metrics, polynomial growth
@article{AIF_2009__59_4_1321_0,
     author = {Sanders, Tom},
     title = {A Fre\u\i man-type theorem for locally compact abelian groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     pages = {1321-1335},
     doi = {10.5802/aif.2465},
     mrnumber = {2566962},
     zbl = {1179.43002},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_4_1321_0}
}
Sanders, Tom. A Freĭman-type theorem for locally compact abelian groups. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1321-1335. doi : 10.5802/aif.2465. http://www.numdam.org/item/AIF_2009__59_4_1321_0/

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