Effective equidistribution of rational points on horocycle sections in $\text{ASL}(2,\mathbb{Z}) \backslash \text{ASL}(2,\mathbb{R}) $

TL;DR

This paper establishes an effective equidistribution result for rational points on expanding horocycle sections in the space of affine special linear group, extending previous work and providing explicit convergence rates.

Contribution

It introduces an effective version of equidistribution for rational points on horocycle sections in ASL(2,R), generalizing prior results to a broader setting.

Findings

Proves effective equidistribution for primitive and non-primitive points.

Extends previous results to the affine special linear group setting.

Provides explicit rates of convergence for equidistribution.

Abstract

In this paper we prove an effective equidistribution result for both primitive and non-primitive points on certain expanding horocycle sections in ASL$(2,\mathbb{Z}) \backslash $ASL$(2,\mathbb{R})$. This provides an effective version of a result recently proven by the author and generalises recent work by Einsiedler, Luethi and Shah who prove similar results in the context of the unit tangent bundle of the modular surface.