Counting saddle connections in a homology class modulo <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol q $\end{document}</tex-math></inline-formula> (with an appendix by Rodolfo Gutiérrez-Romo)

Abstract

We give effective estimates for the number of saddle connections on a translation surface that have length \begin{document}$ \leq L $\end{document} and are in a prescribed homology class modulo \begin{document}$ q $\end{document} . 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.