A Freiman-type theorem for locally compact abelian groups
نویسنده
چکیده
— Suppose that G is a locally compact abelian group with a Haar measure μ. The δ-ball Bδ of a continuous translation invariant pseudo-metric is called d-dimensional if μ(B2δ′ ) 6 2μ(Bδ′ ) for all δ′ ⊂ (0, δ]. We show that if A is a compact symmetric neighborhood of the identity with μ(nA) 6 ndμ(A) for all n > d log d, then A is contained in an O(d log d)-dimensional ball, B, of positive radius in some continuous translation invariant pseudo-metric and μ(B) 6 exp(O(d log d))μ(A). Résumé. — Soit G un groupe abélien localement compact muni d’une mesure de Haar μ. La δ-boule Bδ pour une pseudo-métrique continue et invariante par translation sera dite de dimension d si μ(B2δ′ ) 6 2μ(Bδ′ ) pour tout δ′ ⊂ (0, δ]. Nous montrons que si A est un voisinage compact symétrique de l’identité tel que μ(nA) 6 ndμ(A) pour tout n > d log d, alors A est contenu dans une boule B de dimension O(d log d) et de rayon strictement positif pour une pseudo-métrique continue et invariante par translation ; de plus μ(B) 6 exp(O(d log d))μ(A).
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