JournalslemVol. 66, No. 3/4pp. 259–304

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 nn-gons, the set YR2Y \subset \mathbb{R}^2 of saddle connection holonomy vectors satisfies a quadratic growth estimate {yY:yR}cYR2|\{y \in Y: \|y\|\leq R\}| \sim c_YR^2 , and computed the constant cYc_Y. In 1992 he recorded an observation of Sarnak that gives an error estimate {yY:yR}=cYR2+O(R43)|\{y \in Y: \|y\|\leq R\}| = c_YR^2 + O\bigl(R^{\frac{4}{3}}\bigr) 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 SL(R)\operatorname{SL}(\mathbb{R}) on the plane. In this paper we expose this technique and use it to obtain the following results. For lattices Γ\Gamma with trivial residual spectrum, we recover the error estimate O(R43)O(R^{\frac43}), with a simpler proof. Extending this argument to more general shapes, and using twisted Eisenstein series, for sectors Sα,β={reiθ:r>0,αθα+β}\mathcal{S}_{\alpha,\beta} = \{r e^{\mathbf{i}\theta} : r>0, \alpha \leq \theta \leq \alpha+\beta\} we prove an error estimate

{yY:ySα,β,yR}=cYβ2πR2+Oε(R85).\big|\{y \in Y: y \in \mathcal{S}_{\alpha, \beta}, \|y\| \leq R\}\big| = c_Y\frac{\beta}{2\pi} \, R^2 + O_{\varepsilon} \bigl(R^{\frac{8}{5} } \bigr).

For dilations of smooth star bodies RBψ={reiθ:0rRψ(θ)}R\cdot B_\psi =\{r e^{\mathbf{i} \theta}: 0 \leq r \leq R \psi(\theta)\}, where R>0R>0 and ψ\psi is smooth, we prove an estimate

{yY:yRBψ}=cY,ψR2+Oψ,ε(R127).|\{y \in Y: y \in R \cdot B_{\psi}\}|= c_{Y, \psi} R^2 + O_{\psi, \varepsilon}\bigl(R^{\frac{12}{7} }\bigr).

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