Related papers: Property A and asymptotic dimension

To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and $\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension $\text{dim}(X)$ of $X$,…

We build an example of a metric space with transfinite asymptotic dimension $2\omega$.

The concept of a quasi-metric space arises by relaxing the requirement of the symmetry axiom in the definition of a metric. This small variation alters several structural properties possessed by a standard metric space. This article aims to…

We introduce a probability distribution on $\mathcal{P}([0,1]^d)$, the space of all Borel probability measures on $[0,1]^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero…

We show that if the upper Assouad dimension of the compact set $E\subseteq \mathbb{R}$ is positive, then given any $D>\dim_{A}E$ there is a measure with support $E$ and upper Assouad (or regularity) dimension $D$. Similarly, given any…

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Moreover, we obtain almost…

We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space $(X, d)$ is ultrametric iff the diametrical graph of the metric $d_{\varepsilon}(x,…

We analyze the asymptotics of the dimension of components of the Chow variety as degree increases. By analogy with the divisor case, the main goal is to relate the asymptotic behavior with the positivity of the corresponding cycle classes.…

We first prove that for every metrizable space $X$, for every closed subset $F$ whose complement is zero-dimensional, the space $X$ can be embedded into a product space of the closed subset $F$ and a metrizable zero-dimensional space as a…

Let $(X,d)$ be a metric space. A set $S\subseteq X$ is said to be a $k$-metric generator for $X$ if and only if for any pair of different points $u,v\in X$, there exist at least $k$ points $w_1,w_2, \ldots w_k\in S$ such that $d(u,w_i)\ne…

We define thin and asymptotically scattered metric spaces as asymptotic counterparts of discrete and scattered metric spaces respectively. We characterize asymptotically scattered spaces in terms of prohibited subspaces, and classify thin…

The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the `thickest' and `thinnest' parts of the set. Less extreme versions of these…

Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguished closed set $A \subset X$, one may construct in a functorial way a pointed pseudometric space $\mathcal{D}_\infty(X,A)$ of persistence diagrams equipped with the…

We consider asymptotic dimension of coarse spaces. We analyse coarse structures induced by metrisable compactifications. We calculate asymptotic dimension of coarse cell complexes. We calculate the asymptotic dimension of certain negatively…

We introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent $\alpha\geq 0$ such that for any pair of scales $0<r<R$, any ball of…

Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$.…

The goal of this note is to prove the Half Space Property for RCD(0,N) spaces, namely that if (X,d,m) is a parabolic RCD(0,N) space and $ C \subset X \times \mathbb{R}$ is locally the boundary of a perimeter minimizing set and it is…

A metric space $(X,d)$ is called a $subline$ if every 3-element subset $T$ of $X$ can be written as $T=\{x,y,z\}$ for some points $x,y,z$ such that $d(x,z)=d(x,y)+d(y,z)$. By a classical result of Menger, every subline of cardinality $\ne…

Asymptotic methods for hypothesis testing in high-dimensional data usually require the dimension of the observations to increase to infinity, often with an additional condition on its rate of increase compared to the sample size. On the…

There exists in General Relativity an unambiguous notion of Mass associated to asymptotically flat spacetimes known as the ADM mass. The standard expression for the same is a surface integral over spatial infinity of a linear combination of…