Title: Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the $p$-adic numbers,
Abstract: We exhibit the first explicit examples of Salem sets in $\mathbb{Q}_p$ of
every dimension $0 < \alpha < 1$ by showing that certain sets of
well-approximable $p$-adic numbers are Salem sets. We construct measures
supported on these sets that satisfy essentially optimal Fourier decay and
upper regularity conditions, and we observe that these conditions imply that
the measures satisfy strong Fourier restriction inequalities. We also partially
generalize our results to higher dimensions. Our results extend theorems of
Kaufman, Papadimitropoulos, and Hambrook from the real to the $p$-adic setting.