Effective counting for discrete lattice orbits in the plane via Eisenstein series

  • Claire Burrin

    ETH Zürich, Switzerland
  • Amos Nevo

    Technion - Israel Institute of Technology, Haifa, Israel
  • Rene Rühr

    Weizmann Institute, Rehovot, Israel
  • Barak Weiss

    Tel Aviv University, Israel
Effective counting for discrete lattice orbits in the plane via Eisenstein series cover

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Abstract

In 1989 Veech showed that for the flat surface formed by gluing opposite sides of two regular -gons, the set of saddle connection holonomy vectors satisfies a quadratic growth estimate , and computed the constant . In 1992 he recorded an observation of Sarnak that gives an error estimate in the asymptotics. Both Veech's proof of quadratic growth, and Sarnak's error estimate, rely on the theory of Eisenstein series, and are valid in the wider context of counting points in discrete orbits for the linear action of a lattice in on the plane. In this paper we expose this technique and use it to obtain the following results. For lattices with trivial residual spectrum, we recover the error estimate , with a simpler proof. Extending this argument to more general shapes, and using twisted Eisenstein series, for sectors we prove an error estimate

For dilations of smooth star bodies , where and is smooth, we prove an estimate

Cite this article

Claire Burrin, Amos Nevo, Rene Rühr, Barak Weiss, Effective counting for discrete lattice orbits in the plane via Eisenstein series. Enseign. Math. 66 (2020), no. 3/4, pp. 259–304

DOI 10.4171/LEM/66-3/4-1