On the Inhomogeneous Hall's Ray of Period-One Quadratics

Abstract

For quadratics with period-one negative continued fraction expansions,

X\theta =\frac{1}{ a-{\dfrac{\mathstrut 1}{a-{\dfrac{\mathstrut 1}{a- \cdots }}}}},

we show that the inhomogeneous Lagrange spectrum,

\bL (\theta) :=\bigl\{ \liminf\nolimits_{|n|\rightarrow \infty} |n|@\|n\theta -\gamma\| : \gamma \in \funnyR,\; \gamma \not\in \funnyZ+\theta \funnyZ\bigr\},

contains an inhomogeneous Hall's ray $[0,c(\theta)]$ with $$c(\theta)=\tfrac{1}{4}\bigl(1-O(a^{-1/2})\bigr)\hbox{.}

We describe gaps in the spectrum showing that this is essentially best possible. Pictures of computed spectra are included. Investigating such pictures led us to these results.