The local Hurwitz constant and Diophantine approximation on Hecke groups

Author:
J. Lehner

Journal:
Math. Comp. **55** (1990), 765-781

MSC:
Primary 11J06; Secondary 11F99

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035937-8

MathSciNet review:
1035937

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Abstract: Define the Hecke group by

*c*for which for all -irrationals and infinitely many . Set . We call the Hurwitz constant for . It is known that ,

*q*even; ,

*q*odd. In this paper we prove this result by using -continued fractions, as developed previously by D. Rosen. Write

*q*is odd and , then for at least one of .

**[1]**F. Bagemihl and J. R. McLaughlin,*Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions*, J. Reine Angew. Math.**221**(1966), 146-149. MR**0183999 (32:1475)****[2]**M. Fujiwara,*Bemerkung zur Theorie der Approximation der irrationalen Zahlen durch rationale Zahlen*, Tôhoku Math. J.**14**(1918), 109-115.**[3]**A. Haas and C. Series,*The Hurwitz constant and Diophantine approximation on Hecke groups*, J. London Math. Soc.**34**(1986), 219-234. MR**856507 (87m:11060)****[4]**J. Lehner,*Diophantine approximation on Hecke groups*, Glasgow Math. J.**27**(1985), 117-127. MR**819833 (87e:11079)****[5]**D. Rosen,*A class of continued fractions associated with certain properly discontinuous groups*, Duke Math. J.**21**(1954), 549-564. MR**0065632 (16:458d)****[6]**K. Th. Vahlen,*Ueber Näherungswerte und Kettenbrüche*, J. Reine Angew. Math.**115**(1895), 221-233.

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DOI:
https://doi.org/10.1090/S0025-5718-1990-1035937-8

Article copyright:
© Copyright 1990
American Mathematical Society