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 $m\ge 1$, 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 $$G_m=\{\frac{1}{u+v}:u\in \mathbb{Z}[i],v\in G_{m-1},|u+v|>1\}.$$ We prove that for every $m\ge 1$, $${\dim }_{\mathrm{M}}{H}_m={\dim }_{\mathrm{M}}{G}_m=1.$$ 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 $m\ge 2$.