The local Hurwitz constant and Diophantine approximation on Hecke groups
Author:
J. Lehner
Journal:
Math. Comp. 55 (1990), 765781
MSC:
Primary 11J06; Secondary 11F99
MathSciNet review:
1035937
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Abstract: Define the Hecke group by , . We call the rationals, and the irrationals. The problem we treat here is the approximation of irrationals by rationals. Let be the upper bound of numbers 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 where and are the convergents of the continued fraction for . Then . We call the local Hurwitz constant. In the final section we prove some results on the local Hurwitz constant. For example (Theorem 4), it is shown that if q is odd and , then for at least one of .
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 K. Th. Vahlen, Ueber Näherungswerte und Kettenbrüche, J. Reine Angew. Math. 115 (1895), 221233.
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DOI:
http://dx.doi.org/10.1090/S00255718199010359378
PII:
S 00255718(1990)10359378
Article copyright:
© Copyright 1990
American Mathematical Society
