Badly approximable numbers over imaginary quadratic fields

TL;DR

This paper explores the properties of badly approximable numbers over imaginary quadratic fields, characterizing them via continued fractions and demonstrating their existence on entire circles in the complex plane, with applications to algebraic numbers and Fuchsian groups.

Contribution

It introduces a new characterization of badly approximable numbers over imaginary quadratic fields using continued fractions and proves their existence on circles in the complex plane, both effectively and non-effectively.

Findings

Characterization of badly approximable numbers via bounded partial quotients.

Existence of entire circles in a9 with badly approximable points.

Algebraic numbers of every even degree are badly approximable.

Abstract

We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields $K$ and characterize the "badly approximable" numbers, ($z$ such that there is a $C(z)>0$ with $|z-p/q|\geq C/|q|^2$ for all $p/q\in K$), by boundedness of the partial quotients in the continued fraction expansion of $z$. Applying this algorithm to "tagged" indefinite integral binary Hermitian forms demonstrates the existence of entire circles in $\mathbb{C}$ whose points are badly approximable over $K$, with effective constants. By other methods (the Dani correspondence), we prove the existence of circles of badly approximable numbers over any imaginary quadratic field, with loss of effectivity. Among these badly approximable numbers are algebraic numbers of every even degree over $\mathbb{Q}$, which we characterize. All of the examples we consider are associated with cocompact Fuchsian subgroups of the Bianchi groups $SL_2(\mathcal{O})$, where $\mathcal{O}$ is the ring of integers in an imaginary quadratic field.