On the distribution of primitive subgroups of $\mathbb{Z}^{d}$ of large covolume

TL;DR

This paper investigates the distribution of high-covolume primitive subgroups of integer lattices, extending previous theorems by analyzing their limiting behavior in a geometric space.

Contribution

It establishes the existence and computes the limiting distribution of the images of rank-(d-1) primitive subgroups with large covolume in a specific geometric space, extending prior results.

Findings

Proves the existence of the limiting distribution.

Computes the explicit form of the limiting distribution.

Extends previous theorems by Aka, Einsiedler, and Shapira.

Abstract

We prove existence and compute the limiting distribution of the image of rank-$\left(d-1\right)$ primitive subgroups of $\mathbb{Z}^{d}$ of large covolume in the space $X_{d-1,d}$ of homothety classes of rank-$\left(d-1\right)$ discrete subgroups of $\mathbb{R}^{d}$. This extends a theorem of Aka, Einsiedler and Shapira.