Sweeping out on a set of integers

Abstract

Abstract. Let (X, ®, 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 > 2, then there exists a set A of measure less than /•"' S*_! k~l such that X = U,es T"A. Thus for every infinite set of integers W there exist sets A of arbitrarily small measure such that X = U nniv T"A.