Diophantine approximation for negatively curved manifolds

Sa'ar Hersonsky, Frédéric Paulin

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, the Hurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space.

Original languageEnglish
Pages (from-to)181-226
Number of pages46
JournalMathematische Zeitschrift
Issue number1
StatePublished - 1 Jan 2002

ASJC Scopus subject areas

  • General Mathematics


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