Salem sets with no arithmetic progressions

Abstract

We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ 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 Laba and Pramanik on pseudo-random subsets of the real line which do contain progressions.

11 pages, no figures. v3: typos and minor issues fixed

References in corpus (1)

• On the sharpness of Mockenhaupt's restriction theorem

Cited by in corpus (2)

• On the Fourier analytic structure of the Brownian graph
• Large sets containing no copies of a given infinite sequence