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Farey-type subdivision for organizing quadratic approximants in C

Construct an analogue of the Farey subdivision that organizes quadratic algebraic approximants to complex numbers, analogous to how the Farey tessellation underlies continued fractions and Diophantine approximation on the real line.

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Background

The classical Farey tessellation in the upper half-plane organizes rationals and underlies continued fractions. Schmidt’s complex subdivision and Apollonian packings provide new structures in C, and the notes develop a geometric framework for quadratic approximation, but a direct analogue of the Farey subdivision tailored to quadratic approximants is not yet known.

Finding such a subdivision could yield a combinatorial/dynamical encoding of best quadratic approximations in the complex plane.

References

There are a great many open problems motivated by this perspective. Is there an analog to the Farey subdivision?

— 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”