New bounds for Szemeredi's theorem, II: A new bound for $r_4(N)$

TL;DR

This work advances the pursuit of quantitative bounds for $r_4(N)$ by sharpening the density-increment method through an inverse $U^3$ framework and a robust quadratic-structure theory. The authors show that large $U^3$-norm implies correlation with locally quadratic phases on regular Bohr sets, then build a quadratic Koopman-von Neumann decomposition to obtain substantive density increments on quadratic Bohr sets. By linearising these quadratic structures into long arithmetic progressions, they achieve a significantly tighter bound $r_4(N) \ll N e^{-c\sqrt{\log \log N}}$ for large $N$, setting the stage for the even stronger bound targeted in Part III. The methods integrate finite-field intuition with a careful, graph-like energy increment argument in the integer setting, potentially extending to general finite abelian groups.

Abstract

Define $r_4(N)$ to be the largest cardinality of a set $A$ in $\{1,\dots,N\}$ 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 this paper (part II of a series) we improve this to $r_4(N) \ll N e^{-c\sqrt{\log \log N}}$. In part III of the series we will use a more elaborate argument to improve this to $r_4(N) \ll N(\log N)^{-c}$.

Theorems & Definitions (52)

• Theorem 1.1: Main theorem
• Proposition 1.2: Lack of progressions implies density increment
• Proposition 1.3: Lack of progressions implies density increment
• Remark
• Theorem 3.1: Anomalous number of AP4s implies density increment
• Remark
• Lemma 3.2: $L^1$ controls $\Lambda$
• Lemma 3.3: $U^3$ controls $\Lambda$
• Definition 4.1: Bohr sets
• Example