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.
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Bourgain, J. On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61, 39–72 (1988). https://doi.org/10.1007/BF02776301
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DOI: https://doi.org/10.1007/BF02776301