# Effective counting on translation surfaces

@article{Nevo2017EffectiveCO, title={Effective counting on translation surfaces}, author={Amos Nevo and Ren'e Ruhr and Barak Weiss}, journal={arXiv: Dynamical Systems}, year={2017} }

We prove an effective version of a celebrated result of Eskin and Masur: for any affine invariant manifold of translation surfaces, almost every translation surface has quadratic growth for the saddle connection holonomy vectors, with an effective bound of the error. We also provide effective versions of counting in sectors and in ellipses.

#### 6 Citations

Counting saddle connections in a homology class modulo $q$

- Mathematics
- 2018

We give effective estimates for the number of saddle connections on a translation surface that have length $\leq L$ and are in a prescribed homology class modulo $q$. Our estimates apply to almost…

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

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- 2019

We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of…

Siegel-Veech transforms are in $L^2$

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- 2017

Let $\mathcal{H}$ denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on $\mathbb{R}^2$ is in…

Uniform distribution of saddle connection lengths (with an appendix by Daniel El-Baz and Bingrong Huang)

- MathematicsJournal of Modern Dynamics
- 2019

For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.

ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES

- MathematicsForum of Mathematics, Pi
- 2020

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back…

On convergence of random walks on moduli space

- Mathematics
- 2021

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the n-step…

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