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.