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For $m\geq 2$, we determine the Dirichlet spectrum in $\Rm$ with respect tosimultaneous approximation and the maximum norm as the entire interval $[0,1]$.This complements previous work of several authors, especially Akhunzhanov andMoshchevitin, who considered $m=2$ and Euclidean norm. We construct explicitexamples of real Liouville vectors realizing any value in the unit interval. Inparticular, for positive values, they are neither badly approximable norsingular. Thereby we obtain a constructive proof of the main claim in a recentpaper by Beresnevich, Guan, Marnat, Ram\'irez and Velani, who obtained acountable partition of $[0,1]$ into intervals with each having non-emptyintersection with the Dirichlet spectrum. Our construction is flexible enoughto show that the according set of vectors with prescribed Dirichlet exponenthas large packing dimension and rather large Hausdorff dimension as well. Weestablish a more general result on exact uniform approximation, applicable to awide class of approximating functions. Our constructive proofs are considerablyshorter and less involved than previous work on the topic. By minor twists ofour proof, we infer similar, slightly weaker results when restricting to acertain class of classical fractal sets or other norms. In an Appendix weaddress the situation of a linear form.

Exact uniform approximation and Dirichlet spectrum in dimension at least two