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Density of natural numbers and the Lévy group. (English) Zbl 0658.10065

Let \(\delta\) be the asymptotic density of natural numbers and \({\mathcal F}\) the collection of subsets of \({\mathbb{N}}\) which admit the density. The triple (\({\mathbb{N}},{\mathcal F},\delta)\) may be regarded as an analogue of a probability space. The Lévy group \({\mathcal G}\) is by definition the subgroup of permutations g on \({\mathbb{N}}\) such that \[ \lim_{N\to \infty}(1/N)\quad | \{1\leq n\leq N;\quad g(n)>N\}| =0. \] Then the density \(\delta\) is invariant under \({\mathcal G}\). The main results of this paper are: (i) \(\delta\) is ergodic under \({\mathcal G}\); (ii) \(\delta\) is characterized by the invariance under \({\mathcal G}\) and finite additivity. Moreover, rearrangement of uniformly distributed sequences is discussed in connection with the Lévy group. A preprint quoted in the paper will appear in Nagoya Math. J. under the title “A characterization of the Lévy Laplacian in terms of infinite dimensional rotation groups”.
Reviewer: N.Obata

MSC:

11B05 Density, gaps, topology
20B27 Infinite automorphism groups
60D05 Geometric probability and stochastic geometry
11K06 General theory of distribution modulo \(1\)
20P05 Probabilistic methods in group theory
28D05 Measure-preserving transformations

Citations:

Zbl 0611.60013
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References:

[1] Dunford, N.; Schwartz, J. T., (Linear Operators Part I: General Theory (1967), Interscience: Interscience New York)
[2] Halberstam, H.; Roth, K. F., Sequences I (1966), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0141.04405
[3] Hida, T., (Analysis of Brownian Functionals (1986), IMA, University of Minnesota), Lecture Notes
[4] Hlawka, E., Interpolation analytischer Funktionen auf dem Einheitskreis, (Turán, P., Number Theory and Analysis (1969), Plenum: Plenum New York), 97-118 · Zbl 0212.07301
[5] Kac, M., Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc. (N.S.), 55, 641-665 (1949) · Zbl 0036.30502
[6] Kac, M., (Statistical Independence in Probability Theory, Analysis and Number Theory, Vol. 12 (1959), Wiley: Wiley New York), Carus Monograph · Zbl 0088.10303
[7] Kuipers, L.; Niederreiter, H., (Uniform Distribution of Sequences (1974), Wiley: Wiley New York) · Zbl 0281.10001
[8] Larcher, G., Quantitative rearrangement theorems, Compositio Math., 60, 251-259 (1986) · Zbl 0612.10043
[9] Lévy, P., (Problèmes Concrets d’Analyse Fonctionnelle (1951), Gauthier-Villars: Gauthier-Villars Paris)
[10] Niederreiter, H., A general rearrangement theorem for sequences, Arch. Math. (Basel), 43, 530-534 (1984) · Zbl 0536.54020
[11] Obata, N., A note on certain permutation groups in the infinite dimensional rotation group, Nagoya Math. J., 109, 91-107 (1988) · Zbl 0611.60013
[12] Obata, N., Analysis of the Lévy Laplacian, Soochow J. Math., 14, 115-119 (1988)
[13] Obata, N., The Lévy Laplacian and infinite dimensional rotation groups (1987), preprint
[14] von Neumann, J., Uniformly distributed sequences, Mat. Fiz. Lapok, 32, 32-40 (1925), [Hungarian]
[15] Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77, 313-352 (1916) · JFM 46.0278.06
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