On the spectrum of Diophantine approximation constants

The approximation constant ${\lambda }_k(\zeta )$ is defined as the supremum of real $\eta$ such that $\mathrm{\parallel }{\zeta }^{j}x\mathrm{\parallel }\le x^{-\eta }$ for $1\le j\le k$ has infinitely many integer solutions $x$. Here $\mathrm{\parallel }\mathrm{.}\mathrm{\parallel }$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\lambda }_1(\zeta ),{\lambda }_2(\zeta ),\dots )$ which will lead to various improvements of known results on the individual spectrum of the approximation constants ${\lambda }_k(\zeta )$ as well. In particular, this extends a result by Bugeaud to the case of arbitrary dimension $k$. Concretely, given $k\ge 1$ and $\lambda \ge 1$, we infer {\em explicit} constructions of $\zeta$ in the Cantor set with ${\lambda }_k(\zeta )=\lambda$.