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arXiv · 1510.07596

Salem sets with no arithmetic progressions

Pablo Shmerkin
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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.

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Pablo Shmerkin. 2015-10-30. Salem sets with no arithmetic progressions. https://doi.org/10.1093/imrn%2Frnw097

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