Related papers: Wild high-dimensional Cantor fences in $\mathbb{R}…

In 1994, J.Cobb constructed a tame Cantor set in $\mathbb R^3$ each of whose projections into $2$-planes is one-dimensional. We show that an Antoine's necklace can serve as an example of a Cantor set all of whose projections are…

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particular, we show that a compact subset of $\mathbb{R}^n$ is…

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…

Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all…

Starting with a smooth, non-trivial $n$-dimensional knot $K\subset\bS^{n+2}$, and a beaded $n$-dimensional necklace subordinated to $K$, we construct a wild knot with a Cantor set of wild points (\ie the knot is not locally flat in these…

It is known that if a compact set $E$ in $\mathbb{R}^d$ has Hausdorff dimension greater than $(d+1)/2$, then its $n$-chain distance set $$\Delta^n(E) = \left\{\left(\left|x^1-x^2\right|,\cdots, \left|x^{n}- x^{n+1}\right|\right)\in…

We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A intersect C is nonmeager in C. We also examine variants of this result and…

It is well known that a pair of compact sets in $\mathbb{R}^d$ ($d \in \mathbb{N}$) can be separated by small deformations if the sum of their upper box dimensions is less than $d$. In this paper, we demonstrate that this dimension…

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and…

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which…

We show that an n-dimensional compactum X embeds in R^m, where m>3(n+1)/2, if and only if X x X - \Delta admits an equivariant map to S^{m-1}. In particular, X embeds in R^{2n}, n>3, iff the top power of the (twisted) Euler class of the…

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…

We study the approximately finite-dimensional (AF) $C^*$-algebras that appear as inductive limits of sequences of finite-dimensional $C^*$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra $\mathcal…

We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the…

In 1994, John Cobb asked: given $N>m>k>0$, does there exist a Cantor set in $\mathbb R^N$ such that each of its projections into $m$-planes is exactly $k$-dimensional? Such sets were described for $(N,m,k)=(2,1,1)$ by L.Antoine (1924) and…

For each Cantor set C in $R^{3}$, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in $R^{3}$ with complement having the same fundamental group as the complement of C. This…

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…

In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set, and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct…

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the…

In this paper we consider the long-term behavior of points in ${\mathbb R}$ under iterations of continuous functions. We show that, given any Cantor set $\Lambda^*$ embedded in ${\mathbb R}$, there exists a continuous function $F^*:{\mathbb…