New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r₄(N)

2017/05/04 by Ben Green, Terence Tao, Green, Ben +1 · 4 citations
Mathematics · Engineering · #Limits and Structures in Graph Theory #Analytic Number Theory Research #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1705.01703

Abstract

Define r₄(N) to be the largest cardinality of a set A ⊂ \1,…,N\ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that r₄(N) ≪ N(log log N)^-c for some absolute constant c>0. In 2005, the authors improved this to r₄(N) ≪ N e^{-c√(loglog N)}. In this paper we further improve this to r₄(N) ≪ N(log N)^-c, which appears to be the limit of our methods.

Citations

• On Some Sequences of Integers
• Roth’s theorem on progressions revisited
• On Certain Sets of Integers

Cited by

• Roth-type theorems in additive combinatroics
• Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection
• Pinned patterns and density theorems in \mathbb R^d
• Finding arithmetic progressions in dense sets of integers

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