微型装置的尺寸范围
The range of dimensions of microsets
摘要：
如果存在序列$λ_n\geq 1$和$u_n\in \mathbb{R}^d$使得$(λ_n K + u_n ) \cap [0,1]^d$在Hausdorff度量下收敛于$E$，并且还收敛于$E \cap (0, 1)^d \neq \emptyset$，则称$E$是紧集$K\subset \mathbb{R}^d$的微集.本文的主要结果是：对于非空集$A\subset [0,d]$，存在紧集$K\subset \mathbb{R}^d$，使得由$K$的微集得到的Hausdorff维数集等于$A$当且仅当$A$是解析的且包含其下确界和上确界.这回答了弗雷泽、豪罗伊德、凯恩麦基和于的一个问题。证明了对每个紧集$K\subset \mathbb{R}^d$和非空解析集$A\subset [0,\dim_H K]$，存在一个$K$的紧子集集$\mathcal{C}$，它在Hausdorff度量和$\{\dim_H C: C\in \mathcal{C} \}=A$下是紧的.该证明依赖于随机余维的技术适用于一个适当的耦合的分形维数与世代相关的保留概率。我们还研究了类似的问题，包装和盒尺寸。
英文摘要：
We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $λ_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(λ_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and moreover, $E \cap (0, 1)^d \neq \emptyset$. The main result of the paper is that for a non-empty set $A\subset [0,d]$ there is a compact set $K\subset \mathbb{R}^d$ such that the set of Hausdorff dimensions attained by the microsets of $K$ equals $A$ if and only if $A$ is analytic and contains its infimum and supremum. This answers a question of Fraser, Howroyd, Käenmäki, and Yu. We show that for every compact set $K\subset \mathbb{R}^d$ and non-empty analytic set $A\subset [0,\dim_H K]$ there is a set $\mathcal{C}$ of compact subsets of $K$ which is compact in the Hausdorff metric and $\{\dim_H C: C\in \mathcal{C} \}=A$. The proof relies on the technique of stochastic co-dimension applied for a suitable coupling of fractal percolations with generation dependent retention probabilities. We also examine the analogous problems for packing and box dimensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
