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

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.

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