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Multiple recurrence and convergence for sequences related to the prime numbers

Abstract : For any measure preserving system $(X,\mathcal{X},\mu,T)$ and $A\in\mathcal{X}$ with $\mu(A)>0$, we show that there exist infinitely many primes $p$ such that $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$. Furthermore, we show the existence of the limit in $L^2(\mu)$ of the associated double 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 $p-1$ for some prime $p$.
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https://hal-upec-upem.archives-ouvertes.fr/hal-01267074
Contributor : Bernard Host Connect in order to contact the contributor
Submitted on : Wednesday, February 3, 2016 - 7:37:26 PM
Last modification on : Tuesday, October 19, 2021 - 4:07:15 PM

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  • HAL Id : hal-01267074, version 1

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Nikos Frantzikinakis, Bernard Host, Bryna Kra. Multiple recurrence and convergence for sequences related to the prime numbers. Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2007, 611, pp.131-144. ⟨hal-01267074⟩

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