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

Part of: Classical measure theory Harmonic analysis in several variables Abstract harmonic analysis Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  29 January 2019

Robert Fraser  and
Kyle Hambrook
Robert Fraser
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T1Z2, Canada (rgf@math.ubc.ca)
Kyle Hambrook
Affiliation:
Department of Mathematics and Statistics, San Jose State University, San Jose, CA95192, USA (kyle.hambrook@sjsu.edu)
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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.

Keywords

Hausdorff dimensionFourier dimensionSalem setsFourier restrictionMetric Diophantine approximationp-adic

MSC classification

Primary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Secondary: 11J83: Metric theory 11J61: Approximation in non-Archimedean valuations 28A78: Hausdorff and packing measures 28A80: Fractals 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1265 - 1288
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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