Minkowski geometry of finite Hurwitz continued fractions

Abstract

We study the Minkowski geometry of finite-level sets of Gaussian rationals defined by the lengths of their Hurwitz continued fraction expansions. For each $m\geq 1$, let $H_m$ be the set of points in the fundamental square whose Hurwitz continued fraction expansions have length exactly $m$. We also introduce the relaxed recursive sets defined by $G_0=\{0\}$ and $$G_m=\Big\{\frac{1}{u+v}: u \in\mathbb{Z}[i],\ v\in G_{m-1},\ |u+v|>1 \Big\}.$$ We prove that for every $m\geq 1$, $$\dim_{\rm M} H_m=\dim_{\rm M} G_m=1.$$ We further determine the critical one-dimensional Minkowski content of these sets. We have ${\mathcal M}^1(H_1)={\mathcal M}^1(G_1)=4\pi\log(1+\sqrt{2})$, whereas ${\mathcal M}^1(H_m)={\mathcal M}^1(G_m)=\infty$ for every $m\geq 2$.