Additive structures in sumsets

Abstract

Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+ε}}.

28 pp. Corrected typos. Updated references.

References in corpus (1)

• An inverse theorem for the Gowers U^3 norm

Cited by in corpus (11)

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• An application of a local version of Chang's theorem
• The Kelley--Meka bounds for sets free of three-term arithmetic progressions
• Logarithmic bounds for Roth's theorem via almost-periodicity
• A probabilistic technique for finding almost-periods of convolutions
• On the Decay of the Fourier Transform and Three Term Arithmetic Progressions
• Abundant configurations in sumsets with one dense summand
• Sumsets of dense sets and sparse sets