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# L’Enseignement Mathématique

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**Volume 66, Issue 3, 2020, pp. 259–304**

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

Published online: 2021-05-05

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

Claire Burrin^{[1]}, Amos Nevo

^{[2]}, Rene Rühr and Barak Weiss

^{[3]}(1) ETH Zürich, Switzerland

(2) Technion - Israel Institute of Technology, Haifa, Israel

(3) Tel Aviv University, Israel

In 1989 Veech showed that for the flat surface formed by gluing opposite sides of two regular $n$-gons, the set $Y \subset \mathbb{R}^2$ of saddle connection holonomy vectors satisfies a quadratic growth estimate $|\{y \in Y: \|y\|\leq R\}| \sim c_YR^2 $, and computed the constant $c_Y$. In 1992 he recorded an observation of Sarnak that gives an error estimate $|\{y \in Y: \|y\|\leq R\}| = c_YR^2 + O\bigl(R^{\frac{4}{3}}\bigr)$ 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 $\operatorname{SL}(\mathbb{R})$ on the plane. In this paper we expose this technique and use it to obtain the following results. For lattices $\Gamma$ with trivial residual spectrum, we recover the error estimate $O(R^{\frac43})$, with a simpler proof. Extending this argument to more general shapes, and using twisted Eisenstein series, for sectors $\mathcal{S}_{\alpha,\beta} = \{r e^{\mathbf{i}\theta} : r>0, \alpha \leq \theta \leq \alpha+\beta\}$ we prove an error estimate $$ \big|\{y \in Y: y \in \mathcal{S}_{\alpha, \beta}, \|y\| \leq R\}\big| = c_Y\frac{\beta}{2\pi} \, R^2 + O_{\varepsilon} \bigl(R^{\frac{8}{5} } \bigr). $$ For dilations of smooth star bodies $R\cdot B_\psi =\{r e^{\mathbf{i} \theta}: 0 \leq r \leq R \psi(\theta)\}$, where $R>0$ and $\psi$ is smooth, we prove an estimate $$ |\{y \in Y: y \in R \cdot B_{\psi}\}|= c_{Y, \psi} R^2 + O_{\psi, \varepsilon}\bigl(R^{\frac{12}{7} }\bigr). $$

*Keywords: *Effective counting, discrete lattice orbits, Eisenstein series, Veech surfaces

Burrin Claire, Nevo Amos, Rühr Rene, Weiss Barak: Effective counting for discrete lattice orbits in the plane via Eisenstein series. *Enseign. Math.* 66 (2020), 259-304. doi: 10.4171/LEM/66-3/4-1