Effective counting on translation surfaces

TL;DR

This paper establishes an effective version of a key result in translation surface theory, providing explicit bounds on saddle connection growth and counting in sectors and ellipses for affine invariant manifolds.

Contribution

It introduces effective bounds for saddle connection counts and sector/ellipse counting on translation surfaces within affine invariant manifolds, extending previous asymptotic results.

Findings

Quadratic growth bounds with explicit error terms for saddle connections.

Effective counting formulas in sectors and ellipses.

Generalization of Eskin and Masur's result with quantitative bounds.

Abstract

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.