Badly approximable S-numbers and absolute Schmidt games

Let K be a number field, let S be the set of all normalized, non-conjugate Archimedean valuations of K  , and let KS=∏v∈SKv be the Minkowski space associated with K. We strengthen recent results of [ESK10] and [EGL15] by showing that the set of badly approximable elements of KSKS is HH-absolute winning for a certain family of subspaces of KSKS.