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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8219-1-2016,
author = {Michael Fuchs and Dong Han Kim},
title = {On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {41-57},
zbl = {06574971},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8219-1-2016}
}
Michael Fuchs; Dong Han Kim. On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation. Acta Arithmetica, Tome 172 (2016) pp. 41-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8219-1-2016/