On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems II
Volume 105 / 1993
Studia Mathematica 105 (1993), 207-233
DOI: 10.4064/sm-105-3-207-233
Abstract
We show that if q is greater than one, T is a measure preserving transformation of the measure space (X,β,μ) and f is in $L^{q}(X,β,μ)$ then if ϕ is a non-constant polynomial mapping the natural numbers to themselves, the averages $π_{N}^{-1} ∑_{1≤p≤N} f(T^{ϕ(p)} x) (N = 1, 2, ...) converge μ almost everywhere. Here p runs over the primes and $π_N$ denotes their number in [1, N].
