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A zero-one law for improvements to Dirichlet's Theorem


Authors: Dmitry Kleinbock and Nick Wadleigh
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
Published electronically: January 26, 2018
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Abstract: We give an integrability condition on a function $ \psi $ guaranteeing that for almost all (or almost no) $ x\in \mathbb{R}$, the system $ \vert qx-p\vert< \psi (t)$, $ \vert q\vert<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
Email: kleinboc@brandeis.edu

Nick Wadleigh
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
Email: wadleigh@brandeis.edu

DOI: https://doi.org/10.1090/proc/13685
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
Article copyright: © Copyright 2018 American Mathematical Society

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