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On Kurzweil’s 0-1 law in inhomogeneous Diophantine approximation

Volume 173 / 2016

Michael Fuchs, Dong Han Kim Acta Arithmetica 173 (2016), 41-57 MSC: Primary 11J83, 11K60; Secondary 37E10. DOI: 10.4064/aa8219-1-2016 Published online: 9 March 2016

Abstract

We give a necessary and sufficient condition such that, for almost all $s\in{\mathbb R}$, \[ \|n\theta-s\| \lt \psi(n)\ \quad\text{for infinitely many } n\in{\mathbb N}, \] where $\theta$ is fixed and $\psi(n)$ is a positive, non-increasing sequence. This can be seen as a dual result to classical theorems of Khintchine and Szüsz which dealt with the situation where $s$ is fixed and $\theta$ is random. Moreover, our result contains several earlier ones as special cases: two old theorems of Kurzweil, a theorem of Tseng and a recent result of the second author. We also discuss a similar result (with the same consequences) in the field of formal Laurent series.

Authors

  • Michael FuchsDepartment of Applied Mathematics
    National Chiao Tung University
    Hsinchu 300, Taiwan
    e-mail
  • Dong Han KimDepartment of Mathematics Education
    Dongguk University – Seoul
    Seoul 100-715, Korea
    e-mail

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