复连分数的闵可夫斯基维数与内容
Minkowski geometry of finite Hurwitz continued fractions
AI总结：
研究由赫维茨连分数产生的高斯有理数有限级集及相关松弛递归集的闵可夫斯基几何，证明其维数为\(1\)，并确定了临界一维闵可夫斯基内容，\(m = 1\)时为\(4\pi\log(1+\sqrt{2})\)，\(m\geq2\)时为无穷。
AI中文摘要：
我们研究了由赫维茨连分数产生的高斯有理数有限级集的闵可夫斯基几何。对于每个\(m\geq1\)，令\(H_m\)为基本正方形中赫维茨连分数展开长度恰好为\(m\)的点集。还考虑了由\(G_0 = \{0\}\)和\(G_m=\Big\{\frac{1}{u + v}: u \in\mathbb{Z}[i],\ v\in G_{m - 1},\ |u + v|>1 \Big\}\)定义的松弛递归集。证明了对于每个\(m\geq1\)，\(\dim_{\rm M} H_m=\dim_{\rm M} G_m = 1\)。进一步确定了这些集的临界一维闵可夫斯基内容，\({\mathcal M}^1(H_1)={\mathcal M}^1(G_1)=4\pi\log(1+\sqrt{2})\)，而对于\(m\geq2\)，\({\mathcal M}^1(H_m)={\mathcal M}^1(G_m)=\infty\)。
英文摘要：
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 ${\mathcal M}^1(H_1)={\mathcal M}^1(G_1)=4π\log(1+\sqrt{2})$, whereas ${\mathcal M}^1(H_m)={\mathcal M}^1(G_m)=\infty$ for every $m\geq 2$.
