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On Non-Uniformly Discrete Orbits

Sahar Bashan

TL;DR

The paper addresses how non-uniform lattices in $SL_2(\mathbb{R})$ produce non-uniform discreteness of discrete orbits in $\mathbb{R}^2$ by linking orbit gaps to Diophantine properties of continued fractions. It combines explicit unipotent dynamics with convergents of continued fractions to quantify how proximity of orbit points scales with the quality of rational approximation, proving a 3-point horizontal configuration matching a conjecture of Lelièvre for a sequence $\epsilon_n\to0$. It provides explicit constructions in the golden L setting and a gcd-based weak discreteness framework in $\mathbb{Z}[\phi]$ that yield collinear triples of holonomy vectors with shrinking length. Together, these results advance quantitative understanding of holonomy vectors for non-arithmetic Veech surfaces and suggest directions for extending non-uniform discreteness analyses to other non-uniform lattices.

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 Lelièvre regarding the set of long cylinder holonomy vectors of the "golden L" translation surface: for any $ε>0$, three points of this set can be found on a horizontal line within a distance of $ε$ of each other.

On Non-Uniformly Discrete Orbits

TL;DR

The paper addresses how non-uniform lattices in produce non-uniform discreteness of discrete orbits in by linking orbit gaps to Diophantine properties of continued fractions. It combines explicit unipotent dynamics with convergents of continued fractions to quantify how proximity of orbit points scales with the quality of rational approximation, proving a 3-point horizontal configuration matching a conjecture of Lelièvre for a sequence . It provides explicit constructions in the golden L setting and a gcd-based weak discreteness framework in that yield collinear triples of holonomy vectors with shrinking length. Together, these results advance quantitative understanding of holonomy vectors for non-arithmetic Veech surfaces and suggest directions for extending non-uniform discreteness analyses to other non-uniform lattices.

Abstract

We study the property of uniform discreteness within discrete orbits of non-uniform lattices in , acting on by linear transformations. We provide quantitative consequences of previous results by using Diophantine properties. We give a partial result toward a conjecture of Lelièvre regarding the set of long cylinder holonomy vectors of the "golden L" translation surface: for any , three points of this set can be found on a horizontal line within a distance of of each other.
Paper Structure (3 sections, 7 theorems, 61 equations, 2 figures)

This paper contains 3 sections, 7 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Uniform discreteness and continued fractions
  3. Weak Uniform discreteness

Key Result

Proposition 1.2

For every $0<\epsilon\leq1$, distinct points of $S$ in distance at most $\epsilon$ can be found in the ball $B(0,r)$ where $r=O(\frac{1}{\epsilon^2})$ .

Figures (2)

  • Figure 1: The golden L translation surface, with edge identifications.
  • Figure 2:

Theorems & Definitions (17)

  • Remark 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • proof : Proof of Proposition \ref{['prop']}
  • proof : Proof of Proposition \ref{['con']}
  • proof : Proof of Proposition \ref{['rho']}
  • Theorem 3.1
  • ...and 7 more