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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
##### Keywords:
exponential sums; pointwise ergodic theorem
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##### References:
 [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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