Multiple recurrence and convergence for sequences related to the prime numbers

Nikos Frantzikinakis, Bernard Host, Bryna Kra

Type: Article

Publication Date: 2007-01-26

Citations: 68

DOI: https://doi.org/10.1515/crelle.2007.076

Abstract

For any measure preserving system (X,, μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L2(μ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p.

Locations

CiteSeer X (The Pennsylvania State University) - View - PDF
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Journal für die reine und angewandte Mathematik (Crelles Journal) - View

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