Multiple recurrence and convergence for sequences related to the prime numbers

Nikos Frantzikinakis; Bernard Host; Bryna Kra

doi:10.1515/CRELLE.2007.076

Multiple recurrence and convergence for sequences related to the prime numbers

Nikos Frantzikinakis^*, Bernard Host, Bryna Kra

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

Abstract

For any measure preserving system (X, ,T) and A with (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() 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.

Original language	English (US)
Pages (from-to)	131-144
Number of pages	14
Journal	Journal fur die Reine und Angewandte Mathematik
Issue number	611
DOIs	https://doi.org/10.1515/CRELLE.2007.076
State	Published - Oct 26 2007

Funding

The first author acknowledges the support of NSF grant DMS-0111298 and the third author of NSF grant DMS-0555250.

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1515/CRELLE.2007.076

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abstract = "For any measure preserving system (X, ,T) and A with (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() 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.",

author = "Nikos Frantzikinakis and Bernard Host and Bryna Kra",

note = "Funding Information: The first author acknowledges the support of NSF grant DMS-0111298 and the third author of NSF grant DMS-0555250.",

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AU - Frantzikinakis, Nikos

AU - Host, Bernard

AU - Kra, Bryna

N1 - Funding Information: The first author acknowledges the support of NSF grant DMS-0111298 and the third author of NSF grant DMS-0555250.

PY - 2007/10/26

Y1 - 2007/10/26

N2 - For any measure preserving system (X, ,T) and A with (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() 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.

AB - For any measure preserving system (X, ,T) and A with (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() 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.

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Multiple recurrence and convergence for sequences related to the prime numbers

Abstract

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ASJC Scopus subject areas

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