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2001 On the Inhomogeneous Hall's Ray of Period-One Quadratics
Christopher G. Pinner, Dan Wolczuk
Experiment. Math. 10(4): 487-496 (2001).

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.

Citation

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Christopher G. Pinner. Dan Wolczuk. "On the Inhomogeneous Hall's Ray of Period-One Quadratics." Experiment. Math. 10 (4) 487 - 496, 2001.

Information

Published: 2001
First available in Project Euclid: 26 November 2003

zbMATH: 1028.11043
MathSciNet: MR1881749

Rights: Copyright © 2001 A K Peters, Ltd.

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Vol.10 • No. 4 • 2001
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