The Inhomogeneous Hall's Ray

TL;DR

This paper proves that the inhomogeneous approximation spectrum for any irrational number always contains a continuous interval starting from zero, and in certain cases, the spectrum is exactly a ray, revealing new structural properties.

Contribution

It establishes the existence of a Hall's Ray in the inhomogeneous approximation spectrum for all irrationals and characterizes the spectrum as a ray when partial quotients are unbounded.

Findings

The inhomogeneous spectrum always contains an interval from zero.

When partial quotients are unbounded, the spectrum is a ray.

The spectrum's structure depends on the properties of the irrational number.

Abstract

We show that the inhomogenous approximation spectrum, associated to an irrational number \alpha\ always has a Hall's Ray; that is, there is an \epsilon>0 such that [0,\epsilon) is a subset of the spectrum. In the case when \alpha\ has unbounded partial quotients we show that the spectrum is just a ray.