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.