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.