## Badly approximable numbers over imaginary quadratic fields

### Volume 190 / 2019

#### 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 $C=C(z) \gt 0$ 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, we prove the existence of circles of badly approximable numbers over *any* imaginary quadratic field. 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 ${\rm SL}_2(\mathcal{O})$, where $\mathcal{O}$ is the ring of integers in an imaginary quadratic field.