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

## TL;DR

This paper establishes effective bounds on counting lattice orbit points in the plane using Eisenstein series, improving error estimates for saddle connection holonomies on Veech surfaces.

## Contribution

It introduces new effective bounds for quadratic growth asymptotics of lattice orbits in the plane, utilizing Eisenstein series techniques.

## Key findings

- Improved error bounds for counting in sectors.
- Enhanced estimates for smooth star-shaped domains.
- Application to saddle connection holonomies on Veech surfaces.

## Abstract

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 saddle connection holonomies, for some Veech surfaces. The proof uses Eisenstein series and relies on earlier work of many authors (notably Selberg). Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.