An Effective Slope Gap Distribution for Lattice Surfaces

Abstract

We prove an effective slope gap distribution result first for the square torus and then for general lattice translation surfaces. As a corollary, we obtain a dynamical proof for an effective gap distribution result for the Farey fractions. As an intermediate step, we prove an effective equidistribution result for the intersection points of long horocycles with a particular transversal of the horocycle flow in $\mathrm{SL}_2 (\mathbb R)/\Gamma$ where $\Gamma$ is a lattice.