Pointwise ergodic theorem along the prime numbers
Refining methods from \textit{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 \(L^ q\), \(1\leq q<p\) [\textit{A. Bellow}, Perturbation of a sequence, Adv. Math. (to appear)].
scientific article; zbMATH DE number 4138085