On the maximal ergodic theorem for certain subsets of the integers

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.