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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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