Examples of badly approximable vectors over number fields

TL;DR

This paper constructs examples of badly approximable vectors over number fields by exploring their approximation properties and linking them to compact subspaces associated with quadratic and Hermitian forms, generalizing classical results over rationals.

Contribution

It introduces a method to generate badly approximable vectors over number fields using compact subspaces related to quadratic and Hermitian forms, extending known results from rational numbers.

Findings

Examples of badly approximable vectors over number fields are constructed.

Connections between these vectors and compact subspaces of certain arithmetic quotients are established.

Generalization of quadratic irrational approximation properties to 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 $SL_2(\mathcal{O}_F)\backslash SL_2(F\otimes\mathbb{R})$ 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 $\mathbb{Q}$.