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

Publication Date

6-1-2020

Document Type

Article

Publication Title

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Volume

150

Issue

3

DOI

10.1017/prm.2018.115

First Page

1265

Last Page

1288

Abstract

We exhibit the first explicit examples of Salem sets in Qp of every dimension 0 < α < 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.

Funding Sponsor

Natural Sciences and Engineering Research Council of Canada

Keywords

Fourier dimension, Fourier restriction, Hausdorff dimension, Metric Diophantine approximation, p-adic, Salem sets

Department

Mathematics and Statistics

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