How many points contain homothetic copies in their Hurwitz continued fraction expansion?

TL;DR

This paper demonstrates that the set of complex irrationals with Hurwitz continued fraction expansions contains infinitely many scaled copies of any finite subset of a2^2, and this set has Hausdorff dimension 1, illustrating multidimensional pattern emergence.

Contribution

It establishes that the set of complex irrationals with certain continued fraction properties has Hausdorff dimension 1 and contains all finite patterns as homothetic copies.

Findings

The set of interest has Hausdorff dimension 1.

It contains infinitely many homothetic copies of any finite subset of a2^2.

This provides an example of multidimensional pattern emergence in number theory.

Abstract

We prove that the set of complex irrationals whose partial quotients in their Hurwitz continued fraction expansion are naturally regarded as subsets of $\mathbb Z^2$ and contain infinitely many homothetic copies of any finite subset of $\mathbb Z^2$ is of Hausdorff dimension $1$. Our result provides a clear and concrete example of multidimensional pattern emergence in number-theoretic expansions.