Note on the permutations which preserve Buck's measure density
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 Buck's measure density.
scientific article; zbMATH DE number 5578120