Examples of badly approximable vectors over number fields

Abstract

We consider approximation of vectors $\mathbf{z}\in F\otimes \mathbb{R}\cong {\mathbb{R}}^r\times {\mathbb{C}}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of $S{L}_2({\mathcal{O}}_F)\backslash S{L}_2(F\otimes \mathbb{R})$ naturally associated to (totally indefinite, anisotropic) $F$-rational binary quadratic and Hermitian forms, a generalization of the well-known fact that quadratic irrationals are badly approximable over $\mathbb{Q}$.