发表机构
University of Chinese Academy of Sciences; Tsinghua University; Zhejiang University of Technology; Nankai University（中国科学院大学; 清华大学; 浙江工业大学; 南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨谱族、康托尔集及测度投影的中间覆盖，分析ReLU切线核扰动族的阈值、康托尔集的中间维数熵公式，以及正交投影对测度中间维数的保留性。
AI 中文摘要
我们研究谱族、康托尔集及测度投影的中间覆盖。固定ReLU切线核的两个扰动族具有相同的源阈值，其布朗图像具有相等的盒维阈值和不同的中间阈值，沿规定几何尺度几乎必然成立。对于具有延迟收缩的康托尔集，我们得到该集合及其分数布朗图像的中间维数的熵公式，这些公式给出端点渐近性和第一相变的严格分离定理。振荡变体具有不同的上下中间维数，同时保持Assouad谱。最后，对于非零有限紧支撑测度，若其拟Assouad维数不超过目标维数，则几乎每个正交投影都保留该测度的两个中间维数。
英文摘要
We study intermediate coverings of spectral families, Cantor sets, and projections of measures. Two perturbation families of a fixed ReLU tangent kernel have identical source thresholds. Their Brownian images have equal box thresholds and distinct intermediate thresholds, almost surely along prescribed geometric scales. For Cantor sets with delayed contractions, we obtain entropy formulas for the intermediate dimensions of the sets and their fractional Brownian images. These formulas yield endpoint asymptotics and a sharp separation theorem for the first phase transition. An oscillating variant has distinct lower and upper intermediate dimensions while preserving the Assouad spectrum. Finally, almost every orthogonal projection preserves both intermediate dimensions of a nonzero finite compactly supported measure whose quasi-Assouad dimension does not exceed the target dimension.
Comments59 pages, 1 figures
