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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A zero-one law for improvements to Dirichlet’s Theorem
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by Dmitry Kleinbock and Nick Wadleigh PDF
Proc. Amer. Math. Soc. 146 (2018), 1833-1844 Request permission

Abstract:

We give an integrability condition on a function $\psi$ guaranteeing that for almost all (or almost no) $x\in \mathbb {R}$, the system $|qx-p|< \psi (t)$, $|q|<t$ is solvable in $p\in \mathbb {Z}$, $q\in \mathbb {Z}\smallsetminus \{0\}$ for sufficiently large $t$. Along the way, we characterize such $x$ in terms of the growth of their continued fraction entries, and we establish that Dirichlet’s Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.
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Additional Information
  • Dmitry Kleinbock
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
  • MR Author ID: 338996
  • Email: kleinboc@brandeis.edu
  • Nick Wadleigh
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
  • Email: wadleigh@brandeis.edu
  • Received by editor(s): September 22, 2016
  • Received by editor(s) in revised form: January 24, 2017
  • Published electronically: January 26, 2018
  • Additional Notes: The first-named author was supported by NSF grants DMS-1101320 and DMS-1600814.
  • Communicated by: Nimish Shah
  • © Copyright 2018 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 1833-1844
  • MSC (2010): Primary 11J04, 11J70; Secondary 11J13, 37A17
  • DOI: https://doi.org/10.1090/proc/13685
  • MathSciNet review: 3767339