Analog of the Lagrange spectrum for quadratic approximation in C

Determine whether there is a natural Lagrange-spectrum analogue governing the exponents or constants of best possible approximation of complex numbers by quadratic algebraic numbers, under the hyperbolic-distance/discriminant complexity setup developed for the complex plane.

Background

In the real case, the Lagrange (Markoff) spectrum captures extremal constants governing how well reals can be approximated by rationals and is linked to dynamics on hyperbolic surfaces. The notes develop a geometric framework for quadratic approximation in the complex plane but no spectral structure analogous to the Lagrange spectrum is yet known.

Establishing such a spectrum would parallel classical results and could relate to geodesic dynamics on appropriate hyperbolic orbifolds or to structures visible in the quadratic root-locus images.

References

There are a great many open problems motivated by this perspective. Is there a natural analog to a Lagrange spectrum?

— 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”