A weak type (1,1) inequality for maximal averages over certain sparse sequences

Abstract

Examples are constructed of sparse subsequences of the integers for which the associated maximal averages operator is of weak type (1,1). A consequence, by transference, is that an almost everywhere L^1 -- type ergodic theorem holds for corresponding subsequences of iterates of general measure-preserving transformations. These examples can be constructed so that n_k has growth rate k^m for any prescribed integer power m greater than or equal to 2. Urban and Zienkiewicz have established the same conclusion for other subsequences, which have growth rate k^m for noninteger exponents m sufficiently close to 1; the first novelty here is that the exponent can be arbitrarily large. In contrast, Buczolich and Mauldin have shown that the corresponding conclusion fails to hold if n_k is exactly k^2. The rather simple analysis relies on certain exponential sum bounds of Weil, together with a decomposition of Calderon-Zygmund type in which the exceptional set is defined in terms of the subsequence in question. The subsequences used are closely related to those employed by Rudin in a 1960 paper in which examples of Lambda(p) sets were constructed.