Salem Sets with No Arithmetic Progressions

Abstract

We construct compact Salem sets in R/Z of any dimension (including 1), which do not contain any arithmetic progressions of length 3. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than 1, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension 1. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions) and helps clarify a result of Łaba and Pramanik on pseudo-random subsets of R which do contain progressions.