Multiple recurrence and convergence along the primes

Trevor D. Wooley; Tamar D. Ziegler

doi:10.1353/ajm.2012.0048

Multiple recurrence and convergence along the primes

Trevor D. Wooley
, Tamar D. Ziegler

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

29 Scopus citations

Abstract

Let E ⊂ ℤ be a set of positive upper density. Suppose that P₁,P₂,...,P_k ∈ ℤ[X] are polynomials having zero constant terms. We show that the set E ∩ (E -P₁(p-1)) ∩ ··· ∩ (E - P_k(p-1)) is non-empty for some prime number p. Furthermore, we prove convergence in L² of polynomial multiple averages along the primes.

Original language	English
Pages (from-to)	1705-1732
Number of pages	28
Journal	American Journal of Mathematics
Volume	134
Issue number	6
DOIs	https://doi.org/10.1353/ajm.2012.0048
State	Published - Dec 2012
Externally published	Yes

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title = "Multiple recurrence and convergence along the primes",

abstract = "Let E ⊂ ℤ be a set of positive upper density. Suppose that P1,P2,...,Pk ∈ ℤ[X] are polynomials having zero constant terms. We show that the set E ∩ (E -P1(p-1)) ∩ ··· ∩ (E - Pk(p-1)) is non-empty for some prime number p. Furthermore, we prove convergence in L2 of polynomial multiple averages along the primes.",

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year = "2012",

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AU - Ziegler, Tamar D.

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N2 - Let E ⊂ ℤ be a set of positive upper density. Suppose that P1,P2,...,Pk ∈ ℤ[X] are polynomials having zero constant terms. We show that the set E ∩ (E -P1(p-1)) ∩ ··· ∩ (E - Pk(p-1)) is non-empty for some prime number p. Furthermore, we prove convergence in L2 of polynomial multiple averages along the primes.

AB - Let E ⊂ ℤ be a set of positive upper density. Suppose that P1,P2,...,Pk ∈ ℤ[X] are polynomials having zero constant terms. We show that the set E ∩ (E -P1(p-1)) ∩ ··· ∩ (E - Pk(p-1)) is non-empty for some prime number p. Furthermore, we prove convergence in L2 of polynomial multiple averages along the primes.

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EP - 1732

JO - American Journal of Mathematics

JF - American Journal of Mathematics

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Multiple recurrence and convergence along the primes

Abstract

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