Dirichlet spectrum for simultaneous approximation and a linear form

The Dirichlet spectrum is analogously defined to the famous Lagrange spectrum, but with respect to uniform approximation. We determine the Dirichlet spectrum, for simultaneous approximation as well as for the dual problem of approximation with a linear form, in Euclidean space of any dimension at least 2. It turns out it is as large as it can possibly be, that is it equals the entire interval [0,1]. We also present several refined claims, including on metrical theory and fractal settings. Proof ideas are sketched.