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The Hurwitz constant and diophantine approximation on Hecke groups. (English) Zbl 0605.10018
The authors consider the generalization of Hurwitz’s results that when $$k<\sqrt{5}$$ and $$x$$ is irrational, there must exist infinitely many integers $$p,q$$ such that $$| x-p/q| <1/q^2 k$$ and that $$\sqrt{5}$$ is the best such constant, achieved for critical $$x$$ equivalent to $$(1+\sqrt{5})/2$$. For other irrationals, $$\sqrt{8}$$ is the limit of $$k$$, and the critical values of $$x$$ are equivalent to $$\sqrt{2}$$. This was seen as a property of Fuchsian groups by J. Lehner [Pac. J. Math. 2, 327–333 (1952; Zbl 0047.08202)] and as a property of the Markoff spectrum by the reviewer [Ann. Math. (2) 61, 1–12 (1955; Zbl 0064.04303)] for the modular group and by A. L. Schmidt [J. Reine Angew. Math. 286/287, 341–368 (1976; Zbl 0332.10015)] for other Fuchsian groups.
The authors consider the Hecke group $$G_q: (z\mapsto z+2u, z\mapsto - 1/z)$$ for $$u=\cos \pi /q$$, and the rationals are those generated by the cusps. In Hurwitz’s result, the authors replace $$k<\sqrt{5}$$ by $$k<2 \sqrt{(1-u)^ 2+1}$$ for $$q$$ odd with critical $$x=1-u+\sqrt{(1-u)^ 2+1}$$. When $$q$$ is even, we have $$k<2$$ with critical value $$1$$, but the second minimum is known, i.e., $$2u \sqrt{1-u^2+u^4}$$ with critical value of $$x=(1-u^2+\sqrt{1-u^2+u^4})/u$$.

##### MSC:
 11J04 Homogeneous approximation to one number 11F06 Structure of modular groups and generalizations; arithmetic groups
##### Keywords:
Hurwitz constant; diophantine approximation; Hecke groups
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