Related papers: Excursions on Cantor-like Sets
Ternary coherent configurations are, on the one hand, a special case of multidimensional coherent configurations introduced by L. Babai (2016), and, on the other hand, a natural generalization of association schemes on triples introduced by…
The existence of two different Cantor sets, one of them contained in the set of Liouville numbers and the other one inside the set of Diophantine numbers, is proved. Finally, a necessary and sufficient condition for the existence of a…
We offer a new proof (and review some known proofs) of Cantor's Powerset Theorem (1891), which concerns the non-existence of a surjective function from a set onto its powerset.
This article could be called "theme and variations" on Cantor's celebrated diagonal argument. Given a square nxn tableau T=(a_i^j) on a finite alphabet A, let L be the set of its row-words. The permanent Perm(T) is the set of words…
We introduce several approaches to studying the Cantor-Bendixson decomposition of and the dynamics on the (topological) space of subgroups for various families of countable groups. In particular, we uncover the perfect kernel and the…
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,…
We construct a trigonometric series converging to zero everywhere on a subsequence, with coefficients tending to zero. We show that any such series must satisfy that the subsequence is very sparse, and that the support of the related…
We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a…
We investigate variants of the Erd\H{o}s similarity problem for Cantor sets. We prove that under a mild Hausdorff or packing logarithmic dimension assumption, Cantor sets are not full measure universal, significantly improving the known…
We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by…
We show that Li-Yorke chaos ensures the existence of a scrambled Cantor set.
Let $\ee>0$ and $\fff$ be a family of finite subsets of the Cantor set $\ccc$. Following D. H. Fremlin, we say that $\fff$ is $\ee$-filling over $\ccc$ if $\fff$ is hereditary and for every $F\subseteq\ccc$ finite there exists $G\subseteq…
According to Cantor, a set is a collection into a whole of defined and separate (we shall say distinct) objects. So, a natural question is ``How to treat as `sets' collections of indistinguishable objects?". This is the aim of quasi-set…
For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies…
Harrington and Soare introduced the notion of an n-tardy set. They showed that there is a nonempty $\mathcal{E}$ property Q(A) such that if Q(A) then A is 2-tardy. Since they also showed no 2-tardy set is complete, Harrington and Soare…
We introduce perfect tangles for modular tensor categories. These are intended to generalise the perfect tensors first introduced in the context of a toy model for the AdS/CFT correspondence. We construct perfect tangles for several…
We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped…
It is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. (1) Cantor's first proof of the uncountability of the set of all real numbers does not apply to the set of irrational numbers alone, and,…
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…
Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points…