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

TL;DR

This paper advances the bounds on the size of subsets of natural numbers avoiding four-term arithmetic progressions, achieving a polylogarithmic bound that improves previous results and approaches the limits of current methods.

Contribution

The authors improve the upper bound for r_4(N) to a polylogarithmic form, refining previous exponential and subexponential bounds and approaching the theoretical limit of their techniques.

Findings

r_4(N)  N( log N)^{-c} bound established

Progressed from exponential to polylogarithmic bounds

Results suggest the limit of current methods for bounding 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.