Intermediate coverings, random images, and projections
Abstract
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.
Related
- Prym--Tyurin varieties coming from intermediate coverings of curves
- The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori
- Khintchine dichotomy for images of self-similar measures and almost random walks
- Convergence of Random Products of Projections Under Infinite-Periodic Selections
- Random Projections for Spectral Algorithms: Optimal Rates in Sobolev Norms
BibTeX
@misc{zhao2026intermediate,
title = {{Intermediate coverings, random images, and projections}},
author = {Yunhao Zhao and Yingyu Zhao and Chen Zhou and Shengyang Zang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11680},
eprint = {2610.11680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11680},
}
RIS
TY - UNPB
TI - Intermediate coverings, random images, and projections
AU - Yunhao Zhao
AU - Yingyu Zhao
AU - Chen Zhou
AU - Shengyang Zang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11680
UR - https://arxiv.org/abs/2610.11680
AB - 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.
ER -