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

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 $r_4(N)\ll N(\log N{)}^{-c}$.

26 pages, appeared in Klaus Roth memorial volume. New version addresses minor issues with missing factors of $d^{Cd}$ in Proposition A.9

Cited by in corpus (3)

• New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
• Bounds on Van der Waerden Numbers and Some Related Functions
• On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs