Related papers: Lipschitz images and dimensions
In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar…
We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary…
The Hausdorff distance, the Gromov-Hausdorff, the Fr\'echet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $\inf_\rho…
We show that a set of small box counting dimension can be covered by a H\"older graph from all but a small set of directions, and give sharp bounds for the dimension of the exceptional set, improving a result of B. Hunt and V. Kaloshin. We…
Given $b > 1$ and $y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod 1: n\in\mathbb{N}\}$. Such sets are known to have full Hausdorff dimension, and in…
Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets…
We give sufficient conditions to guarantee that if two self-conformal sets E and F have Lipschitz equivalent subsets of positive measure, then there is a bilipschitz map of E into, or onto, F.
In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the…
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine…
In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of `equi-homogeneity' of a set, which requires a uniformity in the…
Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the…
We show that given any six numbers $r,s,t,u,v,w \in (0,1]$ satisfying $r \leq s \leq \min(t,u) \leq \max(t,u) \leq v \leq w$, it is possible to construct a compact subset of $[0,1]$ with Hausdorff dimension equal to $r$, lower modified box…
We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $\alpha$ with…
In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…
We prove that the Lipschitz free space over a certain type of discrete metric space has the Radon-Nikod\'ym property. We also show that the Lipschitz free space over a complete, locally compact metric space has the Schur or approximation…
The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces. While even approximating the distance up to any practical factor poses an NP-hard problem, its relaxations have proven useful for the problems in…
In this paper, we obtain new bounds for the Hausdorff dimension of planar elliptic measure via the application of quasiconformal mappings, with these bounds depending solely on the ellipticity constant of the matrix. In fact, in our case…
The main result states that a connected conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: the outer and inner metric space structures are metrically…
We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the…
Let G be a Carnot group with homogeneous dimension Q larger than 2 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is Q-1. Moreover we construct self-similar sets…