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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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References:

[1] J. COQUET, Permutations des entiers et répartition des suites, Publ. Math. Orsay (Univ. Paris XI, Orsay), 86-1 (1986), 25-39. · Zbl 0581.10026
[2] V.L. KLEE, Jr., Invariant extensions of linear functionals, Pacific J. Math., 14 (1954), 37-46. · Zbl 0055.10001
[3] L. KUIPERS, H. NIEDERREITER, Uniform distribution of sequences Wiley, New York, 1974. · Zbl 0281.10001
[4] P. LÉVY, Problèms Concretes d’Analyse Fonctionelle, Gauthier-Villars, Paris, 1951. · Zbl 0043.32302
[5] N. OBATA, A note on certain permutation groups in the infinite dimensional rotation groups, Nagoya Math. J., 109 (1988), 91-107. · Zbl 0611.60013
[6] N. OBATA, Density of natural numbers and the Lévy group J. Number Theory, 30 (1988), 288-297. · Zbl 0658.10065
[7] A. PATERSON, Amenability, A.M.S., Providence, 1988. · Zbl 0648.43001
[8] H. RINDLER, Eine Charakterisierung gleichverteilter Folgen, Arch. Math., 32 (1979), 185-188. · Zbl 0409.10036
[9] Q. STOUT, On Levi’s duality between permutations and convergent series J. London Math. Soc., (2) 34 (1986), 67-80. · Zbl 0633.40004
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