Examples of badly approximable vectors over number fields

2018/09/19 by Hines, Robert
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1809.07404

Abstract

We consider approximation of vectors z∈ F⊗ℝ≅ℝ^r×ℂ^s by elements of a number field F and construct examples of badly approximable vectors. These examples come from compact subspaces of SL₂(O_F)\backslash SL₂(F⊗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 ℚ.

Related

• Badly approximable vectors on rational quadratic varieties
• Badly approximable numbers over imaginary quadratic fields
• On badly approximable numbers
• Infinitely badly approximable affine forms
• Badly approximable vectors in affine subspaces: Jarn'ık-type result