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Bulletin of the London Mathematical Society Advance Access published online on September 13, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp074
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© 2009 London Mathematical Society

On the structure of steps of three-term arithmetic progressions in a dense set of integers

P. Candela

Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WB
United Kingdom

Received 26 January 2009.

We use recent results in quadratic Fourier analysis to examine the additive structure of the set of steps (or common differences) of three-term arithmetic progressions in a general subset of [N]={1, 2, ..., N} of fixed positive density. In particular, combining the decomposition results of Gowers and Wolf with the recurrence results of Green and Tao, we show that if A sub [N] has density {alpha} > 0, then, for some positive constant c = c({alpha}), the set of steps of three-term arithmetic progressions in A contains an arithmetic progression of length at least c(log log N)c. This improves on the estimate of shape {Omega}{alpha} (log log log log log N) that one can obtain by a straightforward application of Gowers’ bounds for Szemerédi's theorem.

2000 Mathematics Subject Classification 11P99 (primary), 43A60, 42C99 (secondary).


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