Related papers: Excursions on Cantor-like Sets

We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field…

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set,…

Cameron introduced a bijection between the set of sum-free sets and the set of all zero-one sequences. In this paper, we study the sum-free sets of natural numbers corresponding to certain zero-one sequences which contain the Cantor-like…

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower…

For $\lambda\in(0,1/3]$ let $C_\lambda$ be the middle-$(1-2\lambda)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ \Lambda(t):=\left\{\lambda\in(0,1/3]:…

We consider a generalized Cantor set $E(\omega)$ for an infinite sequence $\omega=(q_n)_{n=1}^{\infty}$ of positive numbers with $0<q_n<1$, and examine the quasiconformal equivalence to the standard middle one-third Cantor set…

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that…

The notion of complex dimension of a one-dimensional Cantor set $C=\bigcap_{n=1}^\infty C_n$ dates back decades. It is defined as the set of poles of the meromorphic $\zeta$-function $\zeta(s)=\sum_{n=1}^{\infty}d_j^s$, where $\Re s>0$, and…

This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…

Below, by space we mean a separable metrizable zero-dimensional space. It is studied when the space can be embedded in a Cantor set while maintaining the algebraic structure. Main results of the work: every space is an open retract of a…

Consider $d$ disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map…

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever…

There exist uniformly quasiregular maps $f:\mathbb{R}^3 \to \mathbb{R}^3$ whose Julia sets are wild Cantor sets.

We propose a theory of chaos for discrete systems, based on their representation in a space of ``binary histories'', $ {\cal B^{\infty}} $. We show that $ {\cal B^{\infty}} $ is a metrizable Cantor set which embeds the attractor $\Lambda$,…

Each topological group $G$ admits a unique universal minimal dynamical system $(M(G),G)$. When $G$ is a non-compact locally compact group the phase space $M(G)$ of this universal system is non-metrizable. There are however topological…

A perfect tensor of order $d$ is a state of four $d$-level systems that is maximally entangled under any bipartition. These objects have attracted considerable attention in quantum information and many-body theory. Perfect tensors…

For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap…

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology…

Perfect tensors are the tensors corresponding to the absolutely maximally entangled states, a special type of quantum states of interest in quantum information theory. We establish a method to compute parameterized families of perfect…

We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of…