Salem sets with no arithmetic progressions

TL;DR

This paper constructs Salem sets in the unit circle of any dimension, including 1, that contain no 3-term arithmetic progressions, contrasting with discrete cases and clarifying prior results on pseudo-random sets.

Contribution

It provides the first explicit construction of Salem sets with no 3-term arithmetic progressions across all dimensions, including the full dimension 1 case.

Findings

Constructed Salem sets of any dimension without 3-term arithmetic progressions

Sets can be Ahlfors regular for dimensions less than 1

Measures can be Frostman for dimension 1

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.