On Kurzweil's 0-1 Law in Inhomogeneous Diophantine Approximation

We give a sufficient and necessary condition such that for almost all $s\in{\mathbb R}$ \[ \|n\theta-s\|<\psi(n)\qquad\text{for infinitely many}\ n\in{\mathbb N}, \] where $\theta$ is fixed and $\psi(n)$ is a positive, non-increasing sequence. This improves upon an old result of Kurzweil and contains several previous results as special cases: two theorems of Kurzweil, a theorem of Tseng and a recent result of the second author. Moreover, we also discuss an analogue of our result in the field of formal Laurent series which has similar consequences.