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 ${\mathrm{SL}}_2(\mathbb{R})/\hat{I}“$ where $\hat{I}“$ is a lattice.

37 pages, 3 figures