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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Explicit Salem sets and applications to metrical Diophantine approximation


Author: Kyle Hambrook
Journal: Trans. Amer. Math. Soc. 371 (2019), 4353-4376
MSC (2010): Primary 11J83, 28A78, 28A80, 42A38, 42B10
DOI: https://doi.org/10.1090/tran/7613
Published electronically: November 5, 2018
MathSciNet review: 3917225
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Abstract: Let $Q$ be an infinite subset of ${\mathbb {Z}}$, let $\Psi : {\mathbb {Z}} \rightarrow [0,\infty )$ be positive on $Q$, and let $\theta \in {\mathbb {R}}$. Define \begin{equation*} E(Q,\Psi ,\theta ) = \{ x \in {\mathbb {R}} : \| q x - \theta \| \leq \Psi (q) \text { for infinitely many $q \in Q$} \}. \end{equation*} We prove a lower bound on the Fourier dimension of $E(Q,\Psi ,\theta )$. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation by determining the Hausdorff dimension of $E(Q,\Psi ,\theta )$ in various cases. We also prove a multidimensional analogue of our result.


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Additional Information

Kyle Hambrook
Affiliation: Department of Mathematics and Statistics, San Jose State University, One Washington Square, San Jose, California 95192
MR Author ID: 952267
ORCID: 0000-0002-0097-4257
Email: kyle.hambrook@sjsu.edu

Received by editor(s): October 14, 2014
Received by editor(s) in revised form: April 9, 2016, and April 13, 2018
Published electronically: November 5, 2018
Article copyright: © Copyright 2018 American Mathematical Society