The Spectrum of $\mathbb{Q}$-Isotropic Binary Quadratic Forms

TL;DR

This paper classifies the spectrum of values for certain rational lattices in the plane, revealing a connection to Markoff numbers and providing a complete list of points above 1/3.

Contribution

It provides a complete characterization of the spectrum of $ ext{Q}$-isotropic binary quadratic forms and links the limit points to Markoff numbers.

Findings

Spectrum points above 1/3 are fully listed.

Limit points correspond to specific Markoff number-based values.

The set of limit points is explicitly described in terms of Markoff numbers.

Abstract

We give a complete list of the points in the spectrum $$\mathcal{Z}=\{\inf_{(x,y)\in\Lambda,xy\neq0}{\left\vert xy\right\vert},\,\text{$\Lambda$ is a unimodular rational lattice of $\mathbb{R}^2$}\}$$ above $\frac{1}{3}.$ We further show that the set of limit points of $\mathcal{Z}$ with values larger than $\frac{1}{3},$ is equal to the set $\{\frac{2m}{\sqrt{9m^2-4}+3m},\text{ where $m$ is a Markoff number}\}$.