Ergodic transformations and sequences of integers
type: Item
instance of: scholarly article
title: Ergodic transformations and sequences of integers
zbMATH Open document ID: 0771.28012
DOI: 10.1007/BF02787185
author: S. Eigen
author: Alireza Hajian
author: Steven Kalikow
published in: Israel Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 11B99
zbMATH DE Number: 63274
zbMATH Keywords: wandering set
zbMATH Keywords: complementing sets
zbMATH Keywords: ergodic measure-preserving transformation
zbMATH Keywords: cutting-stacking construction
reviewed by: Pierre Liardet
MaRDI profile type: Publication
cites work: Q5797130
cites work: Sequences of integers and ergodic transformations
cites work: Ergodic measure preserving transformations of finite type
cites work: Q5614084
cites work: Q5614777
cites work: Addition theorems for sets of integers
full work available at URL: https://doi.org/10.1007/bf02787185
OpenAlex ID: W2031164406
Recommended article: Sequences of integers and ergodic transformations
Recommended article: Ergodic measure preserving transformations of finite type
Recommended article: An ε-free Rohlin lemma
Recommended article: Multiple recurrence of Markov shifts and other infinite measure preserving transformations
Recommended article: Q4691573
Recommended article: Q3146451
Recommended article: Q3705529
Recommended article: Q3341077
Recommended article: Q3015966
description: scientific article; zbMATH DE number 63274