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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
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