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Pointwise ergodic theorem along the prime numbers. (English) Zbl 0695.28007
Refining methods from J. Bourgain [Isr. J. Math. 61, No.1, 39-72 (1988; Zbl 0642.28010); ibid. 73-84 (1988; Zbl 0642.28011)] the author succeeds in proving the following very remarkable theorem:
Let $${\mathbb{P}}$$ be the set of prime numbers, $$(X,\mu,T)$$ a measure- preserving system where $$\mu (X)=1.$$. If $$p>1,$$ then for any $$f\in L^ p(X,\mu)$$ $$S_ n(x)=(1/\pi (n))\sum_{p\leq n,p\in {\mathbb{P}}}f(T^ px)$$ converges for almost every x in X ($$\pi$$ (n) is the number of primes not exceeding n), i.e. the set of prime numbers is a “good” universal set for $$L^ p(X,\mu)$$ for any $$p>1.$$ (Bourgain settled the case $$r>(1+\sqrt{3})/2.)$$
The proof uses deep results primary from number-theory, harmonic analysis and functional analysis. It should be noted that for any $$p>1$$ there exist “good” universal sets for $$L^ p$$, which are not “good” universal for $$L^ q$$, $$1\leq q<p$$ [A. Bellow, Perturbation of a sequence, Adv. Math. (to appear)].
Reviewer: H.Rindler

MSC:
 28D05 Measure-preserving transformations 11N05 Distribution of primes 40H05 Functional analytic methods in summability
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References:
 [1] A. Bellow,Perturbation of a sequence, Advances in Math., to appear. · Zbl 0687.28010 [2] J. Bourgain,On the maximal ergodic theorem for certain subsets of the integers, Isr. J. Math.61 (1988), 39–72. · Zbl 0642.28010 · doi:10.1007/BF02776301 [3] J. Bourgain,On the pointwise ergodic theorem on L p for arithmetic sets, Isr. J. Math.61 (1988), 73–84. · Zbl 0642.28011 · doi:10.1007/BF02776302 [4] J. Bourgain,An approach to pointwise ergodic theorems, GAFA-Seminar 1987, Lecture Notes in Math., Springer-Verlag, Berlin, to appear. [5] A. P. Calderon,Ergodic theory and translation-invariant operators, Proc. Natl. Acad. Sci. U.S.A.59 (1968), 349–353. · Zbl 0185.21806 · doi:10.1073/pnas.59.2.349 [6] G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1971. · Zbl 0020.29201 [7] G. H. Hardy, J. E. Littlewood and G. Pólya,Inequalities, Cambridge University Press, 1952. [8] R. C. Vaughan,The Hardy-Littlewood Method, Cambridge University Press, 1981. · Zbl 0455.10034 [9] A. Zygmund,Trigonometric Series, Cambridge University Press, 1968. · Zbl 0157.38204
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