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On the pointwise ergodic theorem on \(L^ p\) for arithmetic sets. (English) Zbl 0642.28011
In this paper the author extends the according pointwise ergodic theorem for \(L^ 2\)-functions of the paper above to the case \(p>(\sqrt{5+1})/2\), i.e. \(\frac{1}{N}\sum_{n<N}T^{p(n)}f\) converges almost surely for any \(f\in L^ p(X,\mu)\) (p(n) a polynomial with integer coefficients). The method is based on interpolation and the results of the first part. In order to make the proof not too complicated “only” the case of \(p(n)=n^ t\) is explicitely proved.
Reviewer: H.Rindler

MSC:
28D05 Measure-preserving transformations
11L40 Estimates on character sums
42A05 Trigonometric polynomials, inequalities, extremal problems
11K06 General theory of distribution modulo \(1\)
60F99 Limit theorems in probability theory
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[1] J. Bourgain,On the maximal ergodic theorem for certain subsets of the integers, Isr. J. Math.61 (1988), 39–72, this issue. · Zbl 0642.28010 · doi:10.1007/BF02776301
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