宏观中间维数的投影定理

中文摘要 AI 辅助

我们通过组合给定尺度之外所有二进环的覆盖贡献，引入了欧几里得空间中无界子集的下宏观中间维数和上宏观中间维数。对于每个无界Borel集$E\subset\R^d$和$m\in \{1,\ldots,d-1\}$，我们证明了对于几乎所有$m$维线性子空间$V\subset \R^d$，$P_VE$的宏观中间维数等于相应的内在容量剖面。一个全测度集同时适用于两个维数和每个参数$\theta\in[0,1]$。证明结合了覆盖-容量比较与指数间隙、投影覆盖的分数矩估计以及源壳的容量加权保留。这些维数在参数为零时恢复宏观Hausdorff维数，在拟等距下不变，并且在正参数处连续。上宏观中间维数与文献\cite{LXZ}中引入的上离散中间维数一致，而下宏观中间维数可能不同。我们计算了有界基离散数字Moran集的投影维数，包括当基和数字集为常数时获得的自相似情形，并给出了对Lipschitz图和算术和集的应用。

英文摘要

We introduce lower and upper macroscopic intermediate dimensions of unbounded subsets of Euclidean space by combining covering contributions from all dyadic annuli beyond a given scale. For every unbounded Borel set $E\subset\R^d$ and $m\in \{1,\ldots,d-1\}$, we prove that the macroscopic intermediate dimensions of $P_VE$ equal the corresponding intrinsic capacity profiles for almost every $m$-dimensional linear subspaces $V\subset \R^d$. One full-measure set works for both dimensions and every parameter $θ\in[0,1]$. The proof combines a covering--capacity comparison with an exponent gap, fractional-moment estimates for projected coverings, and capacity-weighted retention of source shells. The dimensions recover macroscopic Hausdorff dimension at parameter zero, are invariant under quasi-isometries, and are continuous at positive parameters. The upper macroscopic intermediate dimension agrees with the upper discrete intermediate dimension introduced in \cite{LXZ}, whereas the lower one can differ. We compute the projected dimensions of bounded-base discrete digit Moran sets, including the self-similar case obtained when the bases and digit sets are constant, and give applications to Lipschitz graphs and arithmetic sumsets.

发表机构

• Guangzhou University（广州大学）
• Ningbo University（宁波大学）

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