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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.

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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”