On Non-Uniformly Discrete Orbits

TL;DR

This paper investigates the uniform discreteness of orbits of non-uniform lattices in SL(2,R) acting on R^2, providing quantitative Diophantine results and partial progress on a conjecture related to translation surface holonomy vectors.

Contribution

It offers new quantitative insights into the discreteness properties of orbits and advances understanding of a conjecture concerning holonomy vectors on a specific translation surface.

Findings

Partial result towards Lelièvre's conjecture on holonomy vectors.

Identification of Diophantine conditions affecting orbit discreteness.

Quantitative bounds on the proximity of points in holonomy sets.

Abstract

We study the property of uniform discreteness within discrete orbits of non-uniform lattices in $SL_2(\mathbb{R})$, acting on $\mathbb{R}^2$ by linear transformations. We provide quantitative consequences of previous results by using Diophantine properties. We give a partial result toward a conjecture of Leli\`evre regarding the set of long cylinder holonomy vectors of the "golden L" translation surface: for any $\epsilon>0$, three points of this set can be found on a horizontal line within a distance of $\epsilon$ of each other.