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A Characterization of the Quadratic Irrationals

Published online by Cambridge University Press:  20 November 2018

Tom C. Brown
Tom C. Brown*
Affiliation:
Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A ISA
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Abstract

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Let α be a positive irrational real number, and let fα (n) = [(n + l)α] — [nα] — [α],n > 1, where [x] denotes the greatest integer not exceeding x. It is shown that the sequence fα has a certain 'substitution property' if and only if α is the root of a quadratic equation over the rationals.

Keywords

10L10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 1 , 01 March 1991 , pp. 36 - 41
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bernoulli, J. III, Sur une nouvelle espèce de calcul, Recueil pour les astronomes, Vols. 1, 2, Berlin, 1772.Google Scholar
2. Berstel, J., Mots de Fibonacci, Séminaire d'Informatique Théorique, L. I. T. P. Université Paris VI et VII, Année 1980/81, 57–58.Google Scholar
3. Fraenkel, A. S., Mushkin, M., and Tassa, U., Determination of[n9] by its sequence of differences, Canad. Math. Bull. 21 (1978), 441–446.Google Scholar
4. Karhumaki, J., On cube-free w -words generated by binary morphisms, Discrete Appl. Math. 5(1983), 279– 297.Google Scholar
5. Markoff, A. A., Sur une question de Jean Bernoulli, Math. Ann. 19 (1882), 27–36.Google Scholar
6. Restivo, A., Permutation properties and the Fibonacci semigroup, Semigroup Forum 38 (1989), 337–345.Google Scholar
7. Stolarsky, K. B., Beatty sequences, continued fractions, and certain shift operators, Canad. Math, Bull. 19 (1976), 473–482.Google Scholar
8. Venkov, B. A., Elementary Number Theory, Translated and edited by Helen Alderson, Wolters-Noordhoff, Groningen, 1970,65–68.Google Scholar