随机可数稳定zonotope的极点集的Hausdorff维数
Hausdorff Dimension of the Set of Extreme Points of a Random Countable Stable Zonotope
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中文总结 AI 辅助
该研究确定了随机可数稳定zonotope极点集的Hausdorff维数，证明其几乎必然为$(d-1)\alpha$，并给出了临界Hausdorff测度几乎必然有限的结论，同时阐述了上下界的推导方法。
中文摘要 AI 辅助
设$d\geq2$，$0<\alpha<1$，$\Gamma_k$是$(0,\infty)$上标准泊松过程的相继到达时间。给定独立于$(\Gamma_k)$的独立均匀方向$\varepsilon_k\in S^{d-1}$，我们考虑随机可数稳定zonotope $Z_\alpha=\bigoplus_{k=1}^{\infty}\Gamma_k^{-1/\alpha}[0,\varepsilon_k]$。对于其极点集$\operatorname{ext} Z_\alpha$，我们证明几乎必然有$\dim_H \operatorname{ext} Z_\alpha=(d-1)\alpha$，且临界Hausdorff测度$\mathcal H^{(d-1)\alpha}(\operatorname{ext} Z_\alpha)$几乎必然有限。下界由参数化场的稳定增量的切向非退化性和Frostman能量准则得到；上界则通过构造自适应覆盖得到：在每个尺度上，泊松跳变被分为大跳变和小跳变，大跳变确定有限超平面排列，小跳变的和控制其胞腔像的直径。
英文摘要
Let $d\geq 2$, $0<α<1$, and let $Γ_k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S^{d-1}$, independent of $(Γ_k)$, we consider the random countable stable zonotope $Z_α=\bigoplus_{k=1}^{\infty}Γ_k^{-1/α}[0,\varepsilon_k]$. For its set of extreme points $\operatorname{ext} Z_α$, we prove that almost surely $\dim_H \operatorname{ext} Z_α=(d-1)α$, and that the critical Hausdorff measure $\mathcal H^{(d-1)α}(\operatorname{ext} Z_α)$ is almost surely finite. The lower bound follows from the tangential non-degeneracy of the stable increments of the parametrizing field and Frostman's energy criterion. For the upper bound we construct an adaptive covering: at each scale the Poisson jumps are split into large and small ones, the large jumps determine a finite hyperplane arrangement, and the sum of the small jumps controls the diameters of the images of its cells.
