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Diophantine approximation on hyperbolic Riemann surfaces. (English) Zbl 0593.10028

One fundamental object of interest in diophantine approximation is the quantity
\[ \nu (x)=\inf \{k\in {\mathbb{R}}:\quad | x-p/q| <k/q^ 2\quad \text{for infinitely many integers } p \text{ and } q\} \]
which describes how well a real number \(x\) is approximated by rationals. A classical theorem of A. Markov states that there is a discrete set of values \(\mu_i\) decreasing to \(1/3\) so that if \(\nu(x) > 1/3\) then \(\nu(x)=\mu_i\) for some \(i\). The theorem also provides a good description of the values \(\mu_i\) and the numbers with \(\nu(x) > 1/3\).
Using some straightforward geometrical arguments one can translate the diophantine problem of determining \(\nu(x)\) into the problem of determining the depth of penetration of a certain geodesic on the punctured torus \(M = \mathbb{H}/[\mathrm{SL}(2,\mathbb{Z}), \mathrm{SL}(2,\mathbb{Z})]\) into the cusp. A special case of the results of this paper is that those \(x\) with \(\nu(x) > 1/3\) correspond to endpoints of lifts of simple geodesics on \(M\) to the hyperbolic plane \(\mathbb{H}\).
More generally, let \(M'\) be either a four times punctured sphere or a torus with one puncture or hole (an \(H\)-torus), endowed with a metric of constant negative curvature \(-1\). Let \(S\) denote the set of simple closed geodesics on \(M'\), and \(\bar S\) the set of curves which are limits of curves in \(S\). The main result of this paper is that there are definite bounding curves \(\beta\) surrounding the punctures or hole (horocycles in the case of cusps and constant distance curves for holes) so that a geodesic \(\alpha\) crosses some curve \(\beta\) if and only if \(\alpha\not\in \bar S\), and touches if and only if \(\alpha\in \bar S\setminus S\). Further, exact formulae are given for the position of and the distance from \(\beta\) to a curve \(\alpha\in S\).
The formulae for simple geodesics are derived by dissecting the surface along appropriate lines and applying some hyperbolic trigonometry. The results for non-simple geodesics are derived mainly from topological arguments which show that a non-simple curve necessarily contains a monogon or a bigon bounding a hole or puncture. (Results of A. F. Beardon, J. Lehner and M. Sheingorn [Closed geodesics on a Riemann surface with application to the Markov spectrum, Trans. Am. Math. Soc. 295, 635–647 (1986; Zbl 0597.10024)] closely related to those of this paper, show that \(H\)-tori and three or four times punctured spheres are the only surfaces for which this holds.)
Finally, a symbolic dynamics for geodesics is set up and the particular symbol sequences associated to simple geodesics are identified. In the special case of the surface \(M\) these sequences are translated into information about the continued fraction expansions of endpoints of lifts of geodesics on \(\mathbb{R}\), from which the classical form of Markov’s theorem described above is deduced.
The symbolic approach has also been used by the reviewer [Math. Intell. 7, No. 3, 20–29 (1985; Zbl 0566.10024)] in the classical case to give a complete proof of Markov’s theorem.

MSC:

11J04 Homogeneous approximation to one number
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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