发表机构
Nanjing University of Science and Technology（南京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造一族度量外测度，其临界指数形成介于Hausdorff与填充维数之间的填充谱，并证明其变分公式、乘积不等式及可达轮廓的完整刻画。
AI 中文摘要
我们构造了一族度量外测度，其临界指数构成一个在Hausdorff维数与填充维数之间插值的填充谱。该谱具有可数稳定性、双Lipschitz不变性，并且在填充端点以下的参数范围内是局部Lipschitz连续的，而在填充端点处可能发生跳跃。对于非空紧集，我们证明了用概率测度的窗口局部质量指数表示的变分公式。我们建立了将填充谱与上中间维数配对的不等式，并通过与紧集的乘积获得了逆特征刻画。我们通过单调性、填充端点以下的连续性以及上右Dini导数的尖锐不等式来刻画所有可达的轮廓。每个容许轮廓都由一个紧二进制数字集实现，其均匀数字测度在填充端点以下的每个参数处达到变分上确界，而其互补数字集在每个参数处达到乘积上确界。
英文摘要
We construct a family of metric outer measures whose critical exponents form a packing spectrum interpolating between Hausdorff and packing dimensions. The spectrum is countably stable, bi-Lipschitz invariant, and locally Lipschitz continuous in the parameter below the packing endpoint, where a jump may occur. For nonempty compact sets, we prove a variational formula in terms of windowed local mass exponents of probability measures. We establish product inequalities pairing the packing spectrum with upper intermediate dimensions and obtain a converse characterization by products with compact sets. We characterize all attainable profiles by monotonicity, continuity below the packing endpoint, and a sharp inequality for the upper right Dini derivative. Every admissible profile is realized by a compact binary digit set whose uniform digit measure attains the variational supremum at every parameter below the packing endpoint and whose complementary digit set attains the product supremum at every parameter.
Comments31 pages; comments welcome
