On the maximal ergodic theorem for certain subsets of the integers

Abstract

It is shown that the set of squares {n 2|n=1, 2,…} or, more generally, sets {n t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL 2-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to be “good sequences” forL p-functions,p>1.