Hecke groups and continued fractions
Hecke groups and continued fractions
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赫克群和连分数
DOI：
10.1017/s0004972700012120
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发表时间：
1992
影响因子：
0.7
通讯作者：
Thomas A. Schmidt
中科院分区：
文献类型：
--
作者：
D. Rosen;Thomas A. Schmidt
Hecke群是第一类Fuchsian群。与用普通连分式研究模曲面的测地线类似，第一作者引入的λ-连分式（λF）可以用来研究由Gq确定的曲面上的测地线。本文主要研究与闭测地线相对应的周期连分式，证明了周期连分式的λF的周期近似于经典情形的形式。由此，我们给出：（1）是周期的充要条件;（2）也有这样的周期展开的λq的元素的例子;（3）讨论Pell方程在λq的二次扩张中的解;（4）Gq的丢番图近似的勒让德常数，即γq使得< γq/Q 2，这意味着“约化有限λF形式”的P/Q是真实的α Gq（∞）的收敛.
The Hecke groups are Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for to be periodic; (2) examples of elements of ℚ(λq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the ℚ(λq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α ∉ Gq(∞).