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

• Autorzy: Michael Fuchs , Dong Han Kim
• Języki publikacji: EN

Abstrakty

EN

We give a necessary and sufficient condition such that, for almost all s ∈ ℝ,
||nθ - s|| < ψ(n) for infinitely many n ∈ ℕ,
where θ is fixed and ψ(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 θ 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.

Kategorie tematyczne

• 37E10: Maps of the circle
• 11K60: Diophantine approximation
• 11J83: Metric theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 173
• Numer: 1
• Strony: 41-57

Daty

wydano

2016

Twórcy

autor

Michael Fuchs

• Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

autor

Dong Han Kim

• Department of Mathematics Education, Dongguk University - Seoul, Seoul 100-715, Korea

Identyfikatory

DOI

10.4064/aa8219-1-2016