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

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 subset of [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 (alpha) (log log log log log N) that one can obtain by a straightforward application of Gowers' bounds for Szemeredi's theorem.