# Quantitative error term in the counting problem on Veech wind-tree models

## TL;DR

This paper derives effective asymptotic formulas for counting periodic billiard trajectories in Veech wind-tree models, linking error terms to spectral properties of the Veech group, with explicit estimates for square obstacles.

## Contribution

It provides the first explicit error term estimates for counting periodic trajectories in Veech wind-tree billiards, connecting spectral data to asymptotic counting.

## Key findings

- Asymptotic formulas for periodic billiard trajectories are established.
- Error terms depend explicitly on spectral properties of the Veech group.
- Explicit estimates are provided for obstacles of side length 1/2.

## Abstract

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$-translations) on Veech wind-tree billiards, that is, wind-tree billiards whose underlying compact translation surfaces are Veech surfaces. We show that the error term depends on spectral properties of the Veech group and give explicit estimates in the case when obstacles are squares of side length $1/2$.