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Quadratic Irrationals in the Lower Lagrange Spectrum

Published online by Cambridge University Press:  20 November 2018

Nancy Davis  and
J. R. Kinney
Nancy Davis
Affiliation:
University of Chicago, Chicago, Illinois
J. R. Kinney
Affiliation:
Michigan State University, East Lansing, Michigan
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We let , where the xi are positive integers and i ∈ N, the set of all integers. We define , where where . We let = [0; x1, x2, … ] where

We let

and define

The range of L(ξ) is known as the Lagrange spectrum and the range of M(ξ) as the Markov spectrum. It is known that both are closed and that the Markov spectrum includes the Lagrange spectrum.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 3 , June 1973 , pp. 578 - 584
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Hightower, C. J., The minima of indefinite binary quadratic forms, J. Number Theory 2 (1970), 364–377.Google Scholar
2. Kinney, J. R. and Pitcher, T. S., The Hausdorff-Besicovich dimension of level sets of Perron s modular functions, Trans. Amer. Math. Soc. 124 (1966), 122–130.Google Scholar
3. Kinney, J. R. and Pitcher, T. S., On the lower range of Perron s modular function, Can. J. Math. 21 (1969), 808–816.Google Scholar
4. Kogonija, P., On the connection between the spectra of Lagrange and Markov. II, Tbiliss. Gos. Univ. Trudy Ser. Meh.-Mat. Nauk 102 (1964), 95–104.Google Scholar
5. Kogonija, P., On the connection between the spectra of Lagrange and Markov. III, Tbiliss. Gos. Univ. Trudy Ser. Meh.-Mat. Nauk 102 (1964), 105–113.Google Scholar
6. Kogonija, P., On the connection between the spectra of Lagrange and Markov. IV, Akad. Nauk. Gruzin. S.S.R. Trudy Tbiliss Mat. Inst. Razmadze 29 (1963), 15–35 (1964).Google Scholar
7. Perron, O., Über die Approximation Irrationale Zahlen durch Rationals, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 12 (1921), 3–17.Google Scholar