Ergodic transformations and sequences of integers
Let \(A\) and \(B\) be complementing sets, i.e., infinite subsets of \(\mathbb{N}\) such that \(A\cap B=\{0\}\) and every positive integer has a unique representation in the form \(a+b\) for some \(a\) in \(A\) and \(b\) in \(B\). A subset \(C\) of \(\mathbb{Z}\) is said to be a complement of \(A\) in \(\mathbb{Z}\) if every integer \(n\) has a unique representation in the form \(a+c\) with \(a\in A\) and \(c\in C\). In this paper the authors investigate the particular complementing subset \(A\) of integers consisting of 0 and all finite sums of odd powers of 2. They use the ergodic measure-preserving transformation of finite type \(T\) introduced by \textit{A. Hajian} and \textit{S. Kakutani} [Lect. Notes Math. 160, 45-62 (1970; Zbl 0213.076)] in order to get various complements of \(A\) in \(\mathbb{Z}\). The transformation \(T\) can be defined as an automorphism of \(\mathbb{R}\) (endowed with the Lebesgue measure) using a cutting-stacking construction. Thus, there are subsets \(W\) of same finite Lebesgue measure such that \(\mathbb{R}\) is (up to a null set) the disjoint union of sets \(T^ aW\), \(a\in A\). For such a set \(W\) and any \(x\in W\), it is proved (Proposition 4) that \(C_ x=\{n\in\mathbb{Z}; T^ nx\in W\}\) is a complement of \(A\) and moreover, \(W\) can be chosen such that \(\min\{| n|;n\in C_ x\) and \(n\neq 0\}\) is arbitrarily large (Proposition 6). It is also proved that there exist complements which are not obtained by the above method. Previous results in this direction were obtained by \textit{S. Eigen} and \textit{A. Hajian} [Adv. Math. 73, No. 2, 256-262 (1989; Zbl 0672.28006)] and can be used to generalize the authors discussion for complementing subsets of \(\mathbb{N}\) in general.
ergodic measure-preserving transformation
scientific article; zbMATH DE number 63274
scientific article