  Let $K$ be a number field, let $S$ be the set of all normalized,
non-conjugate Archimedean valuations of $K$, and let $K_{S} = \prod_{v \in S}
K_v$ be the Minkowski space associated with $K$. We strengthen recent results
of \cite{EsdahlKristensen10} and \cite{EinsiedlerGhoshLytle13} by showing that
the set of badly approximable elements of $K_S$ is $\mathcal{H}$-absolute
winning for a certain family of subspaces of $K_{S}$.
