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Szemerédi's Theorem in the Primes

Part of: Sequences and sets

Published online by Cambridge University Press:  19 November 2018

Luka Rimanić  and
Julia Wolf
Luka Rimanić
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK ([email protected]; [email protected])
Julia Wolf
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK ([email protected]; [email protected])
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Abstract

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any subset of the primes of relative density tending to zero sufficiently slowly contains a three-term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of three-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.

Keywords

Green–Tao theoremSzemerédi's theorempseudorandomnessarithmetic combinatorics

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 443 - 457
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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Footnotes

In memory of Kevin Henriot

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