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On the Lower Range of Perron's Modular Function

Published online by Cambridge University Press:  20 November 2018

J. R. Kinney  and
T. S. Pitcher
J. R. Kinney
Affiliation:
Michigan State University, East Lansing, Michigan
T. S. Pitcher
Affiliation:
University of Southern California, Los Angeles, California
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The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequality

is satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will write

for the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will write

It is known (see 6) that

Where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 808 - 816
Copyright
Copyright © Canadian Mathematical Society 1969

References

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2. Kinney, J. R. and Pitcher, T. S., The Hausdorff-Besicovich dimension of the level sets of Perron*s modular function, Trans. Amer. Math. Soc. 124 (1966), 122–130.Google Scholar
3. 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
4. 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
5. Kogonija, P., On the connection between the spectra of Lagrange and Markov. IV, Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat;. Inst. Razmadze 29 (1963), 15–35 (1964).Google Scholar
6. Perron, O., Uber die Approximation Irrationaler Zahlen durch Rationals, S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. 12 (1921), 3–17.Google Scholar