On the Steinhaus property in topological groups |
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Authors: | Hans Weber Enrico Zoli |
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Institution: | a Dipartimento di Matematica ed Informatica, Università degli Studi di Udine, via delle Scienze 206, Udine 33100, Italy b Via Ballanti Graziani 33, Faenza (RA) 48018, Italy |
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Abstract: | Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ0?m?|G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erd?s-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits 2|G|-many μ-nonmeasurable dense subgroups. |
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Keywords: | primary 28C10 54E52 secondary 20K27 22B99 |
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