An analogue of a theorem of Kurzweil

Abstract

A theorem of Kurzweil ('55) on inhomogeneous Diophantine approximation states that if θ is an irrational number, then the following are equivalent: (A) for every decreasing positive function ψ such that , and for almost every , there exist infinitely many such that ‖qθ − s‖ < ψ(q), and (B) θ is badly approximable. This theorem is not true if one adds to condition (A) the hypothesis that the function q ↦ qψ(q) is decreasing. In this paper we find a condition on the continued fraction expansion of θ which is equivalent to the modified version of condition (A). This expands on a recent paper of Kim (2014 Nonlinearity 27 1985–97).