Note on the permutations which preserve Buck's measure density
type: Item
instance of: scholarly article
title: Note on the permutations which preserve Buck's measure density
zbMATH Open document ID: 1220.11012
author: Milan Paštéka
published in: Mediterranean Journal of Mathematics
publication date: 2009-07-14T00:00:00Z
review text: Let \(A\) be a subset of \({\mathbb N}=\{0,1,2,\dots\}.\) We say that \(A\) has asymptotic density \(d(A)\) if the limit \[ d(A)=\lim_{n\rightarrow+\infty}{{|A\cap[1,n]|}\over n} \] exists. In the papers of \textit{N. Obata} [J. Number Theory 30, No. 3, 288--297 (1988; Zbl 0658.10065)], \textit{M. B. Nathanson} and \textit{R. Parikh} [J. Number Theory 124, No. 1, 151--158 (2007; Zbl 1128.11006)], \textit{R. G. Antonini} and \textit{M. Paštéka} [Unif. Distrib. Theory 1, No. 1, 87--109 (2006; Zbl 1146.11005)] the set of permutations \(g: \mathbb N\rightarrow\mathbb N\) ``preserving asymptotic density'', that is, for any \(A\subseteq\mathbb N\), -- (a) \(A\) has asymptotic density if and only if \(g(A)\) has asymptotic density, and -- (b) these two densities are equal, is studied. It is known by the second of the above mentioned papers that (a) implies (b). The author of the paper under review examines the above question for the so called ``Buck's measurability''. For a set \(A\subseteq\mathbb N\), its ``Buck's measure density'' is defined by \[ \mu^*(A)=\inf\left\{{1\over{m_1}}+\cdots+{1\over{m_k}}: A\subseteq \cup_{i=1}^k(r_i+m_i{\mathbb N}),\, k\geq 1 \right\}, \] where \(k\) and \(m_i,1\leq i\leq k,\) are positive integers, \(r_i\in{\mathbb N},\, 1\leq i\leq k,\) and the infimum is taken over all \((m_1,\dots,m_k,r_1,\dots,r_k)\) , \(k\geq 1\) , satisfying the required condition. The concept corresponding to ``having asymptotic density'' is here to be ``Buck's measurable'': the set \(A\) is called ``Buck's measurable'' if and only if \(\mu^*(A)+\mu^*({\mathbb N}\setminus A)=1.\) The author proves some results concerning the set of permutations \(g: \mathbb N\rightarrow\mathbb N\) preserving Buck's measurability and the value of the Buck's measure density. For instance, he proves that this set of permutations is not a group under composition. He also gives an example of a permutation that preserves Buck's measurability but that does not preserve the value of the Buc…
reviewed by: Georges Grekos
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 5578120
zbMATH Keywords: Buck's measure density
zbMATH Keywords: permutation of integers
MaRDI profile type: Publication
full work available at URL: s00009-009-0171-8
OpenAlex ID: W2329703917
DOI: 10.1007/S00009-009-0171-8
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description: scientific article; zbMATH DE number 5578120