Diophantine approximation on conics

TL;DR

This paper studies how well points on conics over the rationals can be approximated by rational points, revealing that the approximation spectra are fully characterized by a specific family of conics with modular symmetry.

Contribution

It introduces general notions of approximability, Lagrange, and Markoff spectra for conics and shows these are fully described by a special family of conics with modular symmetry.

Findings

Spectra of conics can vary widely, but are fully characterized by a specific family.

The special family of conics $ ext{C}_n$ has symmetry under the modular group $ ext{Γ}_0(n)$.

The proof uses the Gross-Lucianovic bijection to relate conics to quaternionic subrings.

Abstract

Given a conic $\mathcal{C}$ over $\mathbb{Q}$, it is natural to ask what real points on $\mathcal{C}$ are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz's theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha-Kim and Cha-Chapman-Gelb-Weiss shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general $\mathcal{C}$ and prove that their behavior is exhausted by the special family of conics $\mathcal{C}_n : XZ = nY^2$, which has symmetry by the modular group $\Gamma_0(n)$ and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of $\operatorname{Mat}^{2\times 2}(\mathbb{Z})$ and classifying invariant lattices in its $2$-dimensional representation.