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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sweeping out on a set of integers


Authors: Martin H. Ellis and Nathaniel A. Friedman
Journal: Proc. Amer. Math. Soc. 72 (1978), 509-512
MSC: Primary 28D05
DOI: https://doi.org/10.1090/S0002-9939-1978-0509244-X
MathSciNet review: 509244
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Abstract: Let $ (X,\mathcal{B},m)$ be a Lebesgue space, $ m(X) = 1$, and let T be an invertible measurable nonsingular aperiodic transformation of X onto X. If S is a set of r integers, $ r \geqslant 2$, then there exists a set A of measure less than $ {r^{ - 1}}\Sigma _{k = 1}^r{k^{ - 1}}$ such that $ X = { \cup _{n \in S}}{T^n}A$. Thus for every infinite set of integers W there exist sets A of arbitrarily small measure such that $ X = { \cup _{n \cap W}}{T^n}A$.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0509244-X
Article copyright: © Copyright 1978 American Mathematical Society