Counting saddle connections in a homology class modulo $q$

TL;DR

This paper provides effective estimates for counting saddle connections within a specific homology class modulo q on translation surfaces, applicable to almost all surfaces in a stratum, enhancing understanding of their geometric and topological properties.

Contribution

It introduces new effective bounds for the number of saddle connections in a given homology class modulo q, applicable to generic translation surfaces in a moduli space stratum.

Findings

Effective estimates for saddle connections in a homology class modulo q

Results hold for almost all translation surfaces in a stratum

Advances understanding of geometric structures in moduli spaces

Abstract

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 all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.