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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 \(x=(1-u^2+\sqrt{1-u^2+u^4})/u\).

11J04 Homogeneous approximation to one number
11F06 Structure of modular groups and generalizations; arithmetic groups
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