Geodesic-flow/continued-fraction framework for quadratic approximation in C

Develop a geodesic-flow or continued-fraction-type theory that systematically produces good approximations of complex numbers by quadratic algebraic numbers, in the setting where approximation quality is measured using the hyperbolic metric on the upper half-plane and the complexity of the quadratic approximants is measured by their discriminant.

Background

The notes reinterpret approximation by algebraic numbers of degree two in geometric terms by identifying quadratic polynomials with points in a Minkowski model and their roots with points in the hyperbolic upper half-plane. Using this perspective, they establish Dirichlet- and Roth-type results with the hyperbolic metric as the distance and discriminant as a complexity measure.

This raises the possibility of a dynamical, Series-style geodesic-cutting or continued-fraction algorithm that would generate convergents approximating complex numbers by quadratic algebraic numbers. No such general framework is currently known.

References

There are a great many open problems motivated by this perspective. Is there a geodesic flow / continued fraction theory for good approximations by quadratics?

— An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits (2412.02050 - Stange, 3 Dec 2024) in Subsection “Open problems,” Section “Diophantine approximation in the complex plane”