Suppose that is a locally compact abelian group with a Haar measure . The -ball of a continuous translation invariant pseudo-metric is called -dimensional if for all . We show that if is a compact symmetric neighborhood of the identity with for all , then is contained in an -dimensional ball, , of positive radius in some continuous translation invariant pseudo-metric and .
Soit un groupe abélien localement compact muni d’une mesure de Haar . La -boule pour une pseudo-métrique continue et invariante par translation sera dite de dimension d si pour tout . Nous montrons que si est un voisinage compact symétrique de l’identité tel que pour tout , alors est contenu dans une boule de dimension et de rayon strictement positif pour une pseudo-métrique continue et invariante par translation ; de plus .
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
@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}, pages = {1321--1335}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {4}, year = {2009}, doi = {10.5802/aif.2465}, zbl = {1179.43002}, mrnumber = {2566962}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2465/} }
TY - JOUR AU - Sanders, Tom TI - A Freĭman-type theorem for locally compact abelian groups JO - Annales de l'Institut Fourier PY - 2009 SP - 1321 EP - 1335 VL - 59 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2465/ DO - 10.5802/aif.2465 LA - en ID - AIF_2009__59_4_1321_0 ER -
%0 Journal Article %A Sanders, Tom %T A Freĭman-type theorem for locally compact abelian groups %J Annales de l'Institut Fourier %D 2009 %P 1321-1335 %V 59 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2465/ %R 10.5802/aif.2465 %G en %F 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://archive.numdam.org/articles/10.5802/aif.2465/
[1] A note on topological groups, Compositio Math., Volume 3 (1936), pp. 427-430 | EuDML | JFM | Numdam | MR | Zbl
[2] On triples in arithmetic progression, Geom. Funct. Anal., Volume 9 (1999) no. 5, pp. 968-984 | DOI | MR | Zbl
[3] A polynomial bound in Freĭman’s theorem, Duke Math. J., Volume 113 (2002) no. 3, pp. 399-419 | DOI | MR | Zbl
[4] A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal., Volume 15 (2005) no. 2, pp. 340-376 | DOI | MR | Zbl
[5] Freĭman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2), Volume 75 (2007) no. 1, pp. 163-175 | DOI | MR | Zbl
[6] A quantitative version of the idempotent theorem in harmonic analysis, Ann. of Math. (2), Volume 168 (2008) no. 3, pp. 1025-1054 | DOI | MR | Zbl
[7] An analog of Freiman’s theorem in groups, Astérisque (1999) no. 258, pp. xv, 323-326 (Structure theory of set addition) | MR | Zbl
[8] Three term arithmetic progressions and sumsets (2007) (To appear) | Zbl
[9] The cardinality of restricted sumsets, J. Number Theory, Volume 96 (2002) no. 1, pp. 48-54 | MR | Zbl
[10] On a generalization of Szemerédi’s theorem, Proc. London Math. Soc. (3), Volume 93 (2006) no. 3, pp. 723-760 | DOI | MR
[11] On sets with small doubling (2007) (arXiv:math/0703309v1) | MR
[12] Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006 | MR | Zbl
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