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The continuous Diophantine approximation mapping of Szekeres

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

Jeffrey C. Lagarias  and
Andrew D. Pollington
Jeffrey C. Lagarias
Affiliation:
AT&T Bell Laboratories, Murray Hill, New Jersey 07974, USA, e-mail: jcl@research.att.com
Andrew D. Pollington
Affiliation:
Brigham Young University, Provo, Utah 84602, USA, e-mail: andy@hamblin.math.byu.edu
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Abstract

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Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.

Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

MSC classification

Secondary: 11J06: Markov and Lagrange spectra and generalizations 11J70: Continued fractions and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 2 , October 1995 , pp. 148 - 172
Copyright
Copyright © Australian Mathematical Society 1995

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