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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
MathSciNet review: 1035937
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Abstract: Define the Hecke group by

$\displaystyle {G_q} = \left\langle {\left( {\begin{array}{*{20}{c}} 1 & {{\lamb... ...{array}{*{20}{c}} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right)} \right\rangle ,$

$ {\lambda _q} = 2\cos \pi /q$, $ q = 3,4, \ldots $. We call $ {G_q}(\infty )$ the $ {G_q}$-rationals, and $ \mathbb{R} - {G_q}(\infty )$ the $ {G_q}$-irrationals. The problem we treat here is the approximation of $ {G_q}$-irrationals by $ {G_q}$-rationals. Let $ M(\alpha )$ be the upper bound of numbers c for which $ \vert\alpha - k/m\vert < 1/c{m^2}$ for all $ {G_q}$-irrationals and infinitely many $ k/m \in {G_q}(\infty )$. Set $ h_q'= {\inf _\alpha }M(\alpha )$. We call $ h_q'$ the Hurwitz constant for $ {G_q}$. It is known that $ h_q'= 2$, q even; $ h_q'= 2{(1 + {(1 - {\lambda _q}/2)^2})^{1/2}}$, q odd. In this paper we prove this result by using $ {\lambda _q}$-continued fractions, as developed previously by D. Rosen. Write

$\displaystyle \alpha - \frac{{{P_{n - 1}}}}{{{Q_{n - 1}}}} = \frac{{{{( - 1)}^{... ...}{\varepsilon _2} \cdots {\varepsilon _n}}}{{{m_{n - 1}}(\alpha )Q_{n - 1}^2}},$

where $ {\varepsilon _i} = \pm 1$ and $ {P_i}/{Q_i}$ are the convergents of the $ {\lambda _q}$-continued fraction for $ \alpha $. Then $ M(\alpha ) = {\overline {\lim } _n}{m_n}(\alpha )$. We call $ {m_n}(\alpha )$ 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 $ {\varepsilon _{n + 1}} = {\varepsilon _{n + 2}} = + 1$, then $ {m_i} \geq {(\lambda _q^2 + 4)^{1/2}} > h_q'$ for at least one of $ i = n - 1,n,n + 1$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1990-1035937-8
PII: S 0025-5718(1990)1035937-8
Article copyright: © Copyright 1990 American Mathematical Society