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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Explicit Salem sets and applications to metrical Diophantine approximation
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by Kyle Hambrook PDF
Trans. Amer. Math. Soc. 371 (2019), 4353-4376 Request permission

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
  • © Copyright 2018 American Mathematical Society
  • 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
  • MathSciNet review: 3917225