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

Michael Fuchs; Dong Han Kim

Acta Arithmetica (2016)

  • Volume: 173, Issue: 1, page 41-57
  • ISSN: 0065-1036

Abstract

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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.

How to cite

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Michael Fuchs, and Dong Han Kim. "On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation." Acta Arithmetica 173.1 (2016): 41-57. <http://eudml.org/doc/279478>.

@article{MichaelFuchs2016,
abstract = { 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. },
author = {Michael Fuchs, Dong Han Kim},
journal = {Acta Arithmetica},
keywords = {metric inhomogeneous Diophantine approximation; 0-1 law irrational rotation; shrinking target property; formal Laurent series},
language = {eng},
number = {1},
pages = {41-57},
title = {On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation},
url = {http://eudml.org/doc/279478},
volume = {173},
year = {2016},
}

TY - JOUR
AU - Michael Fuchs
AU - Dong Han Kim
TI - On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation
JO - Acta Arithmetica
PY - 2016
VL - 173
IS - 1
SP - 41
EP - 57
AB - 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.
LA - eng
KW - metric inhomogeneous Diophantine approximation; 0-1 law irrational rotation; shrinking target property; formal Laurent series
UR - http://eudml.org/doc/279478
ER -

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