Inhomogeneous Diophantine approximation of some Hurwitzian numbers

TL;DR

This paper investigates inhomogeneous Diophantine approximation constants for Hurwitzian numbers, establishing new characterizations, bounds, and proving a conjecture related to these approximation constants.

Contribution

It extends previous work by relating inhomogeneous and homogeneous approximation constants, characterizes pairs with zero constants, and proves a conjecture of Komatsu.

Findings

Characterization of pairs with zero inhomogeneous approximation constant

Bounds on inhomogeneous constants for specific Hurwitzian numbers

Proof of a conjecture of Takao Komatsu

Abstract

We continue the work of Takao Komatsu by considering the inhomogeneous approximation constant L(\theta,\phi) for Hurwitzian numbers \theta, and rationally related \phi(r \theta +m)/n in Q(\theta) +Q. The current work uses a compactness theorem to relate such inhomogeneous constants to the homogeneous approximation constants. Among the new results are: a characterization of such pairs \theta,\phi for which L(\theta,\phi) is zero; consideration of small values of n^2 L(e^{2/s},\phi); and the proof of a conjecture of Komatsu.