Existence of a Hall ray in higher dimensional Lagrange spectra

Abstract

Given any norm $\|\cdot\|_{\mathbb R^2}$ on $\mathbb R^2$, we prove that the Lagrange spectrum for $2$-vectors contains a Hall ray. Equivalently, every sufficiently small $ρ\ge0$ occurs as $\liminf_{q\to\infty}q^{1/2}\min_{p\in\mathbb Z^2}\|qx-p\|_{\mathbb R^2} $ for some $x\in\mathbb R^2\setminus\mathbb Q^2$. Combined with a recent result of Kleinbock \cite{Kleinbock}, this implies that the $2$-dimensional Lagrange spectrum is either a ray, mirroring the behavior for Dirichlet spectra, or a ray preceded by a dense part.