Narrow progressions in the primes

Terence Tao, Tamar Ziegler

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

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 languageEnglish
Title of host publicationAnalytic Number Theory
Subtitle of host publicationIn Honor of Helmut Maier's 60th Birthday
Pages357-379
Number of pages23
ISBN (Electronic)9783319222400
DOIs
StatePublished - 1 Jan 2015

All Science Journal Classification (ASJC) codes

  • General Computer Science
  • General Mathematics

Fingerprint

Dive into the research topics of 'Narrow progressions in the primes'. Together they form a unique fingerprint.

Cite this