Abstract
We give a negative solution to the following counting problem for measure preserving transformations. Forf∈L 1+ (μ), is it true that sup n (N n (f)(x)/n) <∞, μ a.e., where N n (f)(x)=≠{k:f(T k x)/k >1/ n}? One of the consequences is the nonvalidity of J. Bourgain’s Return Time Theorem for pairs of (L 1,L 1) functions.
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This work was completed while this author visited the Departments of Mathematics of University of North Texas and of University of North Carolina at Chapel Hill.
Supported in part by NSF grant DMS 0100078.
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Assani, I., Buczolich, Z. & Mauldin, R.D. AnL 1 counting problem in ergodic theory. J. Anal. Math. 95, 221–241 (2005). https://doi.org/10.1007/BF02791503
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DOI: https://doi.org/10.1007/BF02791503