New bounds for Szemerédi's theorem, III: A polylogarithmic bound for $r_4(N)$

Abstract

Define $r_4(N)$ to be the largest cardinality of a set $A\subset \{1,\dots ,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant $c>0$. In 2005, the authors improved this to \[ r_4(N) \ll N e^{-c\sqrt{\log\log N}}.\] In this paper we further improve this to \[ r_4(N) \ll N(\log N)^{-c},\] which appears to be the limit of our methods.

96 pages, accepted for publication in Mathematika (Special Issue in honour of Klaus Roth). Comments from referees incorporated in v2

References in corpus (2)

• The primes contain arbitrarily long polynomial progressions
• On sets of integers not containing long arithmetic progressions

Cited by in corpus (1)

• A Constructive Lower Bound on Szemerédi's Theorem