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

Abstract

We give a complete list of the points in the spectrum $$\mathcal{Z}=\{\underset{(x,y)\in \Lambda ,xy\ne 0}{\inf }\left|xy\right|,\text{Λ is a unimodular rational lattice of R2}\}$$ 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}\}$.

11 pages