Related papers: Integer Cantor Sets: Arithmetic Combinatorial Prop…
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…
It has been shown that Cantor bubble Julia sets can appear in the dynamics of polynomials and their singular perturbations. In this paper, we present a criterion that guarantees the existence of Cantor bubble Julia sets for certain rational…
We consider products of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is…
We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr…
We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
The paper focuses on some versions of connected dominating set problems: basic problems and multicriteria problems. A literature survey on basic problem formulations and solving approaches is presented. The basic connected dominating set…
The union of a collection of $n$ sets is generally expressed in terms of a characteristic (indicator) function that contains $2^{n}-1$ terms. In this article, a much simpler expression is found that requires the evaluation of $n$ terms…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…
This paper describes a method to find a connection between combinatorial identities and hypergeometric series with a number of examples. Combinatorial identities can often be written as hypergeometric series with unit argument. In a number…
The total Betti numbers of the toric ideal of a simple graph are, in general, highly sensitive to any small change of the graph. In this paper we look at some combinatorial operations that cause total Betti numbers to change in predictable…
For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states…
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…
We show that under natural technical conditions, the sum of a $C^2$ dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions…
In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite…
We use the combinatorial properties of central sets to prove a result about the existence of exponential monochromatic patterns, in the style of Hindman's Finite Sums Theorem. More precisely, we prove that for every finite coloring 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…
Universal representation of geometric patterns of disordered matters is investigated with the aid of general topology. By utilizing the result obtained in the previous study (S. Ohmori, et.al., Phys. Scr. 94, 105213 (2019)) that any…
We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets as a continuous function of the parameters determining these sets. This results extend a previous one of the author and…
We introduce the characteristic numbers and the chromatic polynomial of a tensor. Our approach generalizes and unifies the chromatic polynomial of a graph and of a matroid, characteristic numbers of quadrics in Schubert calculus, Betti…