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

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

25 Citations (Scopus)

Abstract

Let E subset of Z be a set of positive upper density. Suppose that P-1, P-2, ... , P-k is an element of Z[X] are polynomials having zero constant terms. We show that the set E boolean AND (E - p(1)(p - 1)) boolean AND ... (E - P-k(P - 1)) is non-empty for some prime number p. Furthermore, we prove convergence in L-2 of polynomial multiple averages along the primes.

Translated title of the contribution	Multiple recurrence and convergence 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
Publication status	Published - Dec 2012

Bibliographical note

Publisher: The Johns Hopkins University Press

Keywords

DIFFERENCE SETS
ERGODIC AVERAGES
SEQUENCES
THEOREM
POLYNOMIALS
NILSEQUENCES
VARIABLES
NUMBERS

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

abstract = "Let E subset of Z be a set of positive upper density. Suppose that P-1, P-2, ... , P-k is an element of Z[X] are polynomials having zero constant terms. We show that the set E boolean AND (E - p(1)(p - 1)) boolean AND ... (E - P-k(P - 1)) is non-empty for some prime number p. Furthermore, we prove convergence in L-2 of polynomial multiple averages along the primes.",

keywords = "DIFFERENCE SETS, ERGODIC AVERAGES, SEQUENCES, THEOREM, POLYNOMIALS, NILSEQUENCES, VARIABLES, NUMBERS",

author = "Wooley, {Trevor D.} and Ziegler, {Tamar D.}",

note = "Publisher: The Johns Hopkins University Press",

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language = "English",

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pages = "1705--1732",

journal = "American Journal of Mathematics",

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TY - JOUR

T1 - Multiple recurrence and convergence along the primes

AU - Wooley, Trevor D.

AU - Ziegler, Tamar D.

N1 - Publisher: The Johns Hopkins University Press

PY - 2012/12

Y1 - 2012/12

N2 - Let E subset of Z be a set of positive upper density. Suppose that P-1, P-2, ... , P-k is an element of Z[X] are polynomials having zero constant terms. We show that the set E boolean AND (E - p(1)(p - 1)) boolean AND ... (E - P-k(P - 1)) is non-empty for some prime number p. Furthermore, we prove convergence in L-2 of polynomial multiple averages along the primes.

AB - Let E subset of Z be a set of positive upper density. Suppose that P-1, P-2, ... , P-k is an element of Z[X] are polynomials having zero constant terms. We show that the set E boolean AND (E - p(1)(p - 1)) boolean AND ... (E - P-k(P - 1)) is non-empty for some prime number p. Furthermore, we prove convergence in L-2 of polynomial multiple averages along the primes.

KW - DIFFERENCE SETS

KW - ERGODIC AVERAGES

KW - SEQUENCES

KW - THEOREM

KW - POLYNOMIALS

KW - NILSEQUENCES

KW - VARIABLES

KW - NUMBERS

U2 - 10.1353/ajm.2012.0048

DO - 10.1353/ajm.2012.0048

M3 - Article (Academic Journal)

SN - 0002-9327

VL - 134

SP - 1705

EP - 1732

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 6

ER -

Multiple recurrence and convergence along the primes

Abstract

Bibliographical note

Keywords

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