Explicit Salem Sets in $\mathbb{R}^2$

TL;DR

This paper constructs explicit examples of Salem sets in two-dimensional space for all dimensions between 0 and 2, including the first known explicit Salem sets with dimensions between 0 and 1, extending Kaufman's theorem.

Contribution

It provides the first explicit constructions of Salem sets in  for all dimensions, including those less than 1, advancing the understanding of fractal sets in harmonic analysis.

Findings

Explicit Salem sets constructed for all dimensions in 

First explicit Salem sets with dimension between 0 and 1

Extends Kaufman's theorem to explicit examples

Abstract

We construct explicit (i.e., non-random) examples of Salem sets in $\mathbb{R}^2$ of dimension $s$ for every $0 \leq s \leq 2$. In particular, we give the first explicit examples of Salem sets in $\mathbb{R}^2$ of dimension $0 < s < 1$. This extends a theorem of Kaufman.