Related papers: Cantor sets with high-dimensional projections

In \cite{ZW}, the notion of homogenous perfect set as a generalization of Cantor type sets is introduced. Their Hausdorff, lower box-counting, upper box-counting and packing dimensions are studied in \cite{ZW} and \cite{WW}. In this paper,…

The sum theorem and its corollaries are proved for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p-closed sets, using a new notion of relative normality whose topological correspondent is…

This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)- y, x) of the plane R^2, in which F_{ab}(x)= ax if x>0 and bx if x<0. These maps are area-preserving homeomorphisms of the plane that map rays from the…

The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the plane with remarkable combinatorial and geometric properties. However, it doesn't have…

Akin's notion of good measure, introduced to classify measures on Cantor sets has been translated to dimension groups and corresponding traces by Bezuglyi and the author, but emphasizing the simple (minimal dynamical system) case. Here we…

Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective…

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…

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $X\hat{\otimes}_\pi Y$ is strongly subdifferentiable and either $X$ or $Y$ has the metric…

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between…

We show that there exist $(d-1)$ - Ahlfors regular compact sets $E \subset \mathbb{R}^{d}, d\geq 2$ such that for any $t< d-1$, we have \[ \sup_T \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^t}<\infty \] where the supremum is over all tubes $T$…

Let X be a projective variety of complex dimension 3 with log terminal singularities. We prove that every singular Kahler metric on X with bounded Nash entropy and Ricci curvature bounded below induces a compact RCD space homeomorphic to…

Let L be a compact convex set in R^n, and let 1 <= d <= n-1. The set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set…

We give a counterexample to a conjecture of S.E. Morris by showing that there is a compact plane set X such that R(X) has no non-zero, bounded point derivations but such that R(X) is not weakly amenable. We also give an example of a…

We describe wild embeddings of polyhedra into $\mathbb{R}^N$ which show that the answer to the question of B.J. Baker--M. Laidacker (1989) concerning uncountable families of pairwise disjoint compacta can be twofold. The central idea of our…

Inspired by a classical theorem of topological dimension theory, we prove that every geodesic metric space of asymptotic dimension $n$ containing a bi-infinite geodesic can be coarsely separated by a subset $S$ of asymptotic dimension equal…

We answer a question of Banakh, Jab\l{}o\'nska and Jab\l{}o\'nski by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is…

A. Girand has constructed an explicit two-parameter family of flat connections over the complex projective plane $\mathbb{P}^2$. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and…

Let $\gamma\in(0;\frac{1}{2}),\tau\geq 1$ and define the "$\gamma,\tau$ Diophantine set" as: $$D_{\gamma,\tau}:=\{\alpha\in (0;1): ||q\alpha||\geq\frac{\gamma}{q^{\tau}}\quad\forall q\in\Bbb{N}\},\qquad||x||:=\inf_{p\in\Bbb{Z}}|x-p|. $$ In…

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $\delta$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the…

We show that for any dimension t>2(1+alpha K)/(1+K) there exists a compact set E of dimension t and a function alpha-Holder continuous on the plane, which is K-quasiregular only outside of E. To do this, we construct an explicit…