On the spectrum of Diophantine approximation constants

The approximation constant $\lambda_{k}(\zeta)$ is defined as the supremum of real $\eta$ such that $\Vert \zeta^{j}x\Vert\leq x^{-\eta}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $(\lambda_{1}(\zeta),\lambda_{2}(\zeta),\ldots)$ 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 $k\geq 1$ and $\lambda\geq 1$, we infer {\em explicit} constructions of $\zeta$ in the Cantor set with $\lambda_{k}(\zeta)=\lambda$.