Narrow progressions in the primes

Terence Tao*, Tamar Ziegler

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

4 Scopus citations

Abstract

In a previous paper of the authors, we showed that for any polynomials P1…Pk ε Z[m] with P1(0) = … = Pk(0) and any subset A of the primes in [N] = {1, …, N} of relative density at least φ > 0, one can find a "polynomial progression" a+P1(r), …, a+Pk(r) in A with 0 <|r|≤No(1), if N is sufficiently large depending on k, P1, …, Pk and φ. In this paper we shorten the size of this progression to 0 <|r|≤logLN, where L depends on k, P1, …, Pk and φ. In the linear case Pi =(i-1)m, we can take L independent of φ. The main new ingredient is the use of the densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions.

Original languageAmerican English
Title of host publicationAnalytic Number Theory
Subtitle of host publicationIn Honor of Helmut Maier's 60th Birthday
PublisherSpringer International Publishing
Pages357-379
Number of pages23
ISBN (Electronic)9783319222400
ISBN (Print)9783319222394
DOIs
StatePublished - 1 Jan 2015

Bibliographical note

Publisher Copyright:
© Springer International Publishing Switzerland 2015.

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