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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

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
Full Text: DOI
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