On the Inhomogeneous Hall's Ray of Period-One Quadratics - Christopher G. Pinner; Dan Wolczuk (Experimental Mathematics, 2001)
For quadratics with period-one negative continued fraction expansions, Xθ =\frac{1}{ a-{\dfrac{\mathstrut 1}{a-{\dfrac{\mathstrut 1}{a- … }}}}}, we show that the inhomogeneous Lagrange spectrum, \bL (θ) :=\bigl\{ \liminf\nolimits_{|n|→ ∞} |n|@\|nθ -γ\| : γ \in \funnyR,\; γ \not\in \funnyZ+θ \funnyZ\bigr\}, contains an inhomogeneous Hall's ray [0,c(θ)] with $$c(θ)=\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.