Related papers: From Lipschitz embedding to Lipschitz equivalence …

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar…

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.

The higher dimensional Frobenius problem was introduced by a preceding paper [Fan, Rao and Zhang, Higher dimensional Frobenius problem: maximal saturated cones, growth function and rigidity, Preprint 2014]. %the higher dimensional Frobenius…

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that…

We consider a class of homogeneous self-similar sets with complete overlaps and give a sufficient condition for the Lipschitz equivalence between members in this class.

In this paper, we study a class $\mathcal{A}(\lambda ,n,m)$ of self-similar sets with $m$ exact overlaps generated by $n$ similitudes of the same ratio $ \lambda $. We obtain a necessary condition for a self-similar set in…

Given a $k$-self similar set $X\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.

In this paper we provide an up-to-date survey on the study of Lipschitz equivalence of self-similar sets. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets. A…

We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb R}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient…

Let $K=2^\mathbb{N}$ be the Cantor set, let $\mathcal{M}$ be the set of all metrics $d$ on $K$ that give its usual (product) topology, and equip $\mathcal{M}$ with the topology of uniform convergence, where the metrics are regarded as…

We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets…

Let $M$ be a separable metric space. We say that $f=(f_n):M\to c_0$ is a good-$\lambda$-embedding if, whenever $x,y\in M$, $x\ne y$ implies $d(x,y)\le\Vert f(x)-f(y)\Vert$ and, for each $n$, $Lip(f_n)<\lambda$, where $Lip(f_n)$ denotes the…

We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary,…

This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic…

Let $E, F\subset {\Bbb R}^d$ be two self-similar sets, and suppose that $F$ can be affinely embedded into $E$. Under the assumption that $E$ is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between…

The Erd\H{o}s similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences.…

A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large pieces on which it is bi-Lipschitz?…

We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are…

Suppose A is an open subset of a Carnot group G, where G has a discrete analogue, and H is another Carnot group. We show that a Lipschitz function from A to H whose image has positive Hausdorff measure in the appropriate dimension is…