
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 of any dimension (including ) which do not contain any arithmetic progressions of length . Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than , and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension . 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.
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