On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm

Abstract

and M−(θ, φ) = lim inf q→+∞ q‖qθ + φ‖ = lim inf q→−∞ |q|‖qθ − φ‖. ThenM(θ, φ) = min(M+(θ, φ),M−(θ, φ)). These notations are introduced by Cusick, Rockett and Szusz [2].M(θ, φ) orM+(θ, φ) has been treated by Cassels [1], Descombes [3], Sos [9], Cusick et al. [2] and the author [5] by using several algorithms for inhomogeneous Diophantine approximation in which φ is expressed by the continued fraction expansion of θ. However, it is not easy to evaluate M(θ, φ) if it exists for any given pair of θ and φ. In this paper we establish the relationship between M(θ, φ) and the algorithm of Nishioka, Shiokawa and Tamura. Indeed, this was hinted at in [5] but has not been proved yet. If we use this result, we can find the