On the spectrum of Diophantine approximation constants

Abstract

The approximation constant $λ_{k}(ζ)$ is defined as the supremum of real $η$ such that $\Vert ζ^{j}x\Vert\leq x^{-η}$ 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 $(λ_{1}(ζ),λ_{2}(ζ),\ldots)$ which will lead to various improvements of known results on the individual spectrum of the approximation constants $λ_{k}(ζ)$ as well. In particular, this extends a result by Bugeaud to the case of arbitrary dimension $k$. Concretely, given $k\geq 1$ and $λ\geq 1$, we infer {\em explicit} constructions of $ζ$ in the Cantor set with $λ_{k}(ζ)=λ$.