单位区间内齐次Moran集的 magnitude
Magnitude of homogeneous Moran sets in the unit interval
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中文总结 AI 辅助
本文研究单位区间内齐次Moran集的 magnitude 维数，证明其与对应集合的欧氏盒维数一致，并得到常压缩比自相似情形下 magnitude 的渐近式，补充了Willerton的相关结果。
中文摘要 AI 辅助
Magnitude（记为$\boldsymbol{\text{Mag}(X)}$）是紧度量空间的实值不变量，其大尺度增长反映空间的几何性质。Willerton证明，对于紧齐次黎曼流形$X$，$\text{Mag}(tX)$的增长形如$t^{\boldsymbol{\text{dim }X}}$，其主导渐近项中包含$X$的体积。本文研究配备欧氏度量$d$和编码超度量$d_u$的齐次Moran康托集$E$，记$E_u=(E,d_u)$，证明$\text{Mag}(tE_u)$的上、下增长指数（称为$E_u$的 magnitude 维数）分别与$E$的上、下欧氏盒维数一致。在常压缩比$r$的自相似情形下，当$t\to\boldsymbol{\text{∞}}$时，得到$\text{Mag}(tE_u)=t^s/\boldsymbol{\text{ṕ}}(\boldsymbol{\text{log }}t)+o(t^s)$，其中$s$是$E$的豪斯多夫维数，$\boldsymbol{\text{ṕ}}$是周期为$\boldsymbol{-\text{log }r}$的正光滑函数；主导系数$1/\boldsymbol{\text{ṕ}}$的调和平均为$\boldsymbol{m\boldsymbol{\text{log }}m/((m-1)\boldsymbol{\text{Γ}}(s+1))}$，给出了Willerton主导阶渐近式的分形类似，其系数是对数周期的而非常数。
英文摘要
Magnitude, denoted by $\operatorname{Mag}(X)$, is a real-valued invariant of compact metric spaces whose large-scale growth reflects their geometry. Willerton showed that, for a compact homogeneous Riemannian manifold $X$, $\operatorname{Mag}(tX)$ grows like $t^{\dim X}$, with the volume of $X$ appearing in its leading asymptotic terms. We study a homogeneous Moran Cantor set $E$ equipped with the Euclidean metric $d$ and with its coding ultrametric $d_u$, writing $E_u=(E,d_u)$. We prove that the upper and lower growth exponents of $\operatorname{Mag}(tE_u)$, called the magnitude dimensions of $E_u$, coincide respectively with the upper and lower Euclidean box dimensions of $E$. In the self-similar case with constant contraction ratio $r$, we obtain $\operatorname{Mag}(tE_u)=t^s/\widetilde{p}(\log t)+o(t^s)$ as $t\to\infty$, where $s$ is the Hausdorff dimension of $E$ and $\widetilde{p}$ is a positive smooth function of period $-\log r$. The harmonic mean of the leading coefficient $1/\widetilde{p}$ is $m\log m/((m-1)Γ(s+1))$, giving a fractal analogue of Willerton's leading-order asymptotics with a log-periodic, rather than constant, coefficient.
