Sweeping out on a set of integers

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 $X = { \cup _{n \cap W}}{T^n}A$.