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 ⊂ [ N ] has density α > 0, then, for some positive constant c = c (α), 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 Ω α (log log log log log N ) that one can obtain by a straightforward application of Gowers' bounds for Szemerédi's theorem.