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

By Ben Joseph Green, Terence Tao. 2006

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Define r_4(N) to be the largest cardinality of a set A in {1,...,N} which does not contain four elements in arithmetic progression. In 1998 Gowers proved that r_4(N) 0. In this paper (part II of a series) we improve this to r_4(N) << 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) << N(log N)^{-c}.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/abf42c4af5c83844d4ceede3db115a044334869a