Existence of a Hall ray in higher dimensional Lagrange spectra

Abstract

Given any norm $\Vert \cdot {\Vert }_{{\mathbb{R}}^2}$ on ${\mathbb{R}}^2$, we prove that the Lagrange spectrum for $2$-vectors contains a Hall ray. Equivalently, every sufficiently small $\rho \ge 0$ occurs as ${\mathrm{liminf}}_{q\to \infty }{q}^{1/2}{\min }_{p\in {\mathbb{Z}}^2}\Vert qx-p{\Vert }_{{\mathbb{R}}^2}$ 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.

25 pages,0 figures