Obata, Nobuaki Density of natural numbers and the Lévy group. (English) Zbl 0658.10065 J. Number Theory 30, No. 3, 288-297 (1988). 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 Cited in 3 ReviewsCited in 12 Documents 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 Keywords:probability measure; asymptotic density; Lévy group; invariant; ergodic; rearrangement of uniformly distributed sequences Citations:Zbl 0611.60013 PDFBibTeX XMLCite \textit{N. Obata}, J. Number Theory 30, No. 3, 288--297 (1988; Zbl 0658.10065) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.