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Badly approximable numbers over imaginary quadratic fields

Volume 190 / 2019

Robert Hines Acta Arithmetica 190 (2019), 101-125 MSC: Primary 11J70; Secondary 11F06. DOI: 10.4064/aa170810-18-10 Published online: 24 June 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.

Authors

  • Robert HinesDepartment of Mathematics
    University of Colorado
    Campus Box 395
    Boulder, CO 80309-0395, U.S.A.
    e-mail

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