Badly approximable $S$-numbers and absolute Schmidt games

TL;DR

This paper proves that the set of badly approximable elements in the Minkowski space of a number field is large in the sense of absolute winning, extending recent results in Diophantine approximation.

Contribution

It strengthens previous results by showing that badly approximable elements form an absolute winning set in the context of number fields and Minkowski spaces.

Findings

Set of badly approximable elements is $ ext{H}$-absolute winning.

Extends results of Esdahl Kristensen and Einsiedler et al.

Provides new insights into Diophantine approximation in number fields.

Abstract

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 $K_{S}$.