An Effective Slope Gap Distribution for Lattice Surfaces

TL;DR

This paper establishes an effective distribution of slope gaps for lattice surfaces, extending to general cases and providing a dynamical proof for Farey fractions, with implications for horocycle flow equidistribution.

Contribution

It introduces an effective slope gap distribution for lattice surfaces and connects it to horocycle flow dynamics and Farey fractions, advancing understanding of these distributions.

Findings

Effective slope gap distribution for square torus

Extension to general lattice translation surfaces

Dynamical proof for Farey fractions gap distribution

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.