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Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences. (English) Zbl 0735.11004
Summary: We are interested in permutations preserving certain distribution properties of sequences. In particular we consider \(\mu\)-uniformly distributed sequences on a compact metric space \(X\), 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of \(Aut(\mathbb{N})\) leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group \({\mathcal G}\). We show that \({\mathcal G}\) is big in the sense that the Cesàro mean is characterized by its invariance under the Lévy group. As a result, any \({\mathcal G}\)-invariant positive normalized linear functional on \(\ell^ \infty(\mathbb{N})\) is an extension of Cesàro means. Finally we prove that there exist \({\mathcal G}\)-invariant extensions of Cesàro mean to all of \(\ell^ \infty(\mathbb{N})\).

MSC:
11B05 Density, gaps, topology
11K06 General theory of distribution modulo \(1\)
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
20B27 Infinite automorphism groups
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