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

TL;DR

This work proves that the set of complex irrationals whose Hurwitz continued fraction digits c_n(x) in Z^2 contain infinitely many homothetic copies of any finite subset of Z^2 has Hausdorff dimension 1, matching the ambient constraint set E. The authors construct an explicit infinite iterated function system from the Hurwitz expansion to build a seed set with dimension arbitrarily close to 1, then modify it by inserting large integer squares to force the desired multidimensional patterns without sacrificing dimensionality. The lower bound is established via a detailed mass distribution argument on a carefully crafted seed, while the upper bound follows from existing dimension estimates. The result exemplifies multidimensional pattern emergence in a concrete number theoretic expansion and hints at wider applicability to other expansions.

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.

Figures (1)

• Figure 1: The shaded region indicates $\bigcup\{U_{i}\cap U\colon i\in\mathbb D_1\setminus\mathbb D_2\}$.

Theorems & Definitions (21)

• Theorem 1.1
• Lemma 2.1: NT2
• Lemma 2.2: NT2
• Lemma 2.3
• proof
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof