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Diophantine Approximation and Horocyclic Groups

  • R. A. Rankin (a1)

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1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality

(1)

has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

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References

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1. Ford, L. R., A geometrical proof of a theorem of Hurwitz, Proc. Edinburgh Math. Soc, 85 (1917), 5965.
2. Ford, L. R., Automorphic functions (New York, 1929).
3. Petersson, H., Zur analytischen Théorie der Grenzkreisgruppen I, Math. Ann., 15 (1938), 2367.
4. Rankin, R. A., On horocyclic groups, Proc. London Math. Soc. (3), 4 (1954), 219234.
5. Scott, W. T., Approximation to real irrationals by certain classes of rational fractions, Bull. Amer. Math. Soc, 46 (1940), 124–29.
6. Tornheim, L., Approximation to irrationals by classes of rational numbers, Proc. Amer. Math. Soc, 6 (1955), 260264.
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Diophantine Approximation and Horocyclic Groups

  • R. A. Rankin (a1)

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