The local Hurwitz constant and Diophantine approximation on Hecke groups

Abstract:

Define the Hecke group by \[ {G_q} = \left \langle {\left ( {\begin {array}{*{20}{c}} 1 & {{\lambda _q}} \\ 0 & 1 \\ \end {array} } \right ),\left ( {\begin {array}{*{20}{c}} 0 & { - 1} \\ 1 & 0 \\ \end {array} } \right )} \right \rangle ,\] ${\lambda _q} = 2\cos \pi /q$, $q = 3,4, \ldots$. 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 $|\alpha - k/m| < 1/c{m^2}$ 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 \[ \alpha - \frac {{{P_{n - 1}}}}{{{Q_{n - 1}}}} = \frac {{{{( - 1)}^{n - 1}}{\varepsilon _1}{\varepsilon _2} \cdots {\varepsilon _n}}}{{{m_{n - 1}}(\alpha )Q_{n - 1}^2}},\] 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 $i = n - 1,n,n + 1$.