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arXiv · 2404.16810

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

Giorgos Kotsovolis
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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}\}$.

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Giorgos Kotsovolis. 2024-04-25. The Spectrum of $\mathbb{Q}$-Isotropic Binary Quadratic Forms. https://arxiv.org/abs/2404.16810

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