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Diophantine approximations with restrained denominators. Balance condition on decay and growth rates

Part of: Diophantine approximation, transcendental number theory Harmonic analysis in one variable

Published online by Cambridge University Press:  29 August 2025

Volodymyr Pavlenkov Open the ORCID record for Volodymyr Pavlenkov [Opens in a new window]  and
Evgeniy Zorin Open the ORCID record for Evgeniy Zorin [Opens in a new window]
Volodymyr Pavlenkov
Affiliation:
Department of Mathematics, University of York, Heslington, York, England Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine (pavlenkovvolodymyr@gmail.com)
Evgeniy Zorin*
Affiliation:
Department of Mathematics, University of York, Heslington, York, England (evgeniy.zorin@york.ac.uk)
*
*Corresponding author.
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Abstract

We strengthen known results on Diophantine approximation with restricted denominators by presenting a new quantitative Schmidt-type theorem that applies to denominators growing much more slowly than in previous works. In particular, we can handle sequences of denominators with polynomial growth and Rajchmann measures exhibiting arbitrary slow decay, allowing several applications. For instance, our results yield non-trivial lower bounds on the Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers (each with restricted denominators) and enable the construction of Salem subsets of well-approximable numbers of arbitrary Hausdorff dimension.

Keywords

Diophantine approximationsHausdorff dimensionMetric theory of Diophantine approximationInhomogeneous Diophantine approximationFractal Diophantine approximationSchmidt-type theoremRestraint denominatorsRajchman measure

MSC classification

Primary: 11J83: Metric theory
Secondary: 42A61: Probabilistic methods

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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