Intermediate coverings, random images, and projections

Yunhao Zhao; Yingyu Zhao; Chen Zhou; Shengyang Zang

Home/Archive/arXiv

arXiv · 2610.11680

Intermediate coverings, random images, and projections

Yunhao Zhao·Yingyu Zhao·Chen Zhou·Shengyang Zang

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.MG math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yunhao Zhao, Yingyu Zhao, Chen Zhou, Shengyang Zang. 2026-10-08. Intermediate coverings, random images, and projections. https://arxiv.org/abs/2610.11680

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Non-separable arrangements of convex bodies revisited

A problem posed by Erdős in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane intersecting their convex hull also intersects at least one member of the family. Recent work has focused on minimal coverings of NS-families consisting of positive homothetic convex bodies. In this paper, we strengthen these results by establishing their stability analogues and analyze weakly non-separable families of convex polytopes. We also examine weakly $k$-impassable families of convex $d$-polytopes for $0<k<d-1$ and study tightness of NS-lattice arrangements.

math.MG↗

arXiv2026

Approximating the Euclidean Ball by Polytopes: Surprising Rates in High Dimensions

We determine the sharp dimensional dependence of the asymptotic approximation error for the Euclidean unit ball $B_2^d\subset\mathbb{R}^d$ by polytopes with a prescribed number of vertices in the relative symmetric difference metric. Specifically, we prove that $$\lim_{N\to\infty}N^{2/(d-1)}\inf_{P\in P_d(N)}\frac{\operatorname{vol}_d(P\triangle B_2^d)}{\operatorname{vol}_d(B_2^d)}=Θ(d^{-1/2}),$$ where $P_d(N)$ consists of polytopes with at most $N$ vertices and no containment restriction. Equivalently, the classical Laguerre-Delone triangulation numbers satisfy $\operatorname{ldel}_{d-1}=Θ(d^{-1/2})$. This improves the lower bound $Ω(d^{-1})$ of Böröczky (2000) and the upper bound $O(1)$ of Ludwig, Schütt and Werner (2006). In contrast, the corresponding asymptotic coefficient for approximation with a prescribed number of facets has order one, as shown by Kur (2020). We also establish non-asymptotic bounds showing that, for every fixed $\varepsilon\in(0,1)$, the ball admits an $\varepsilon$-approximation with at most $\exp(C(\varepsilon)\sqrt d)$ vertices, whereas approximating the hypercube to the same accuracy requires at least $\exp(c(\varepsilon)d)$ vertices. Consequently, the ball does not maximize the optimal relative approximation error among convex bodies, even up to an absolute multiplicative constant at the same vertex budget. Thus Macbeath's theorem (1951) on the extremality of the ball for inscribed approximation does not extend to polytopes in arbitrary position.

math.MG↗

arXiv2026

An isometric embedding of the space of non-increasing sequences with non-negative terms into the Gromov-Hausdorff space

The Gromov-Hausdorff space is a certain metric space induced by the Gromov-Hausdorff convergence, and we study an isometric embedding into this space. While the subspace of non-negative real numbers in the Euclidean space admits an isometric embedding, we prove that the subspace of $\ell^\infty$ consisting of non-increasing sequences with non-negative terms can be isometrically embedded into the Gromov-Hausdorff space.

math.MG↗