Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the $p$-adic numbers

TL;DR

This paper constructs explicit Salem sets in the $p$-adic numbers with various dimensions, demonstrating their Fourier decay properties and implications for Fourier restriction, extending real number results to the $p$-adic context.

Contribution

It provides the first explicit examples of Salem sets in $Q_p$ for all dimensions between 0 and 1, with measures satisfying optimal Fourier decay and restriction properties.

Findings

Constructed explicit Salem sets in $Q_p$ for all dimensions 0<α<1.

Established measures with optimal Fourier decay on these sets.

Extended real number Salem set results to the $p$-adic setting.

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.