# Multiple recurrence and convergence for sequences related to the prime numbers

@inproceedings{Frantzikinakis2006MultipleRA, title={Multiple recurrence and convergence for sequences related to the prime numbers}, author={Nikos Frantzikinakis and Bernard Host and Bryna Kra}, year={2006} }

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 L 2(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…

## 59 Citations

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We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these…

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We introduce the notion of “E-ergodicity” of a measure-preserving dynamical system (where E is a subset of N). We show that given an E-ergodic system T and aperiodic system S, T can be sped up to…

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Let P denote the set of primes. For a fixed dimension d, CookMagyar-Titichetrakun, Tao-Ziegler and Fox-Zhao independently proved that any subset of positive relative density of P contains an…

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