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

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.