Density of the multidimensional Lagrange spectrum

The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(qα,\mathbb{Z})$ where $α$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One