Israel Journal of Mathematics

, Volume 61, Issue 1, pp 39–72

On the maximal ergodic theorem for certain subsets of the integers

  • J. Bourgain

DOI: 10.1007/BF02776301

Cite this article as:
Bourgain, J. Israel J. Math. (1988) 61: 39. doi:10.1007/BF02776301


It is shown that the set of squares {n2|n=1, 2,…} or, more generally, sets {nt|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL2-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” forLp-functions,p>1.

Copyright information

© The Weizmann Science Press of Israel 1988

Authors and Affiliations

  • J. Bourgain
    • 1
  1. 1.IHESBures-sur-YvetteFrance

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