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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

28D05 Measure-preserving transformations
11N05 Distribution of primes
40H05 Functional analytic methods in summability
Full Text: DOI
[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.
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[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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