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

Michael Fuchs ¹   ; Dong Han Kim ²

¹ Department of Applied Mathematics National Chiao Tung University Hsinchu 300, Taiwan
² Department of Mathematics Education Dongguk University – Seoul Seoul 100-715, Korea

Acta Arithmetica, Tome 173 (2016) no. 1, pp. 41-57

Résumé

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.

Détail

DOI : 10.4064/aa8219-1-2016

Keywords: necessary sufficient condition almost mathbb theta s psi quad text infinitely many mathbb where theta fixed psi positive non increasing sequence seen dual result classical theorems khintchine which dealt situation where fixed theta random moreover result contains several earlier special cases old theorems kurzweil theorem tseng recent result second author discuss similar result consequences field formal laurent series