Related papers: Isolated Diophantine numbers
When working in NF, [1] there is a sense that there are more non-Cantorian sets than Cantorian sets. But it is not that immediate result as one expects, since they are externally equinumerous, and the qualification "Cantorian" is not…
A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…
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 consider 1-forms on, so called, essentially isolated determinantal singularities (a natural generalization of isolated ones), show how to define analogues of the Poincar\'e--Hopf index for them, and describe relations between these…
We introduce a diophantine property of a log canonical algebra, and use it to describe the restriction of a log canonical algebra of general type to a log canonical center of codimension one.
A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm…
This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.
We obtain some new results on the topology of unary definable sets in densely ordered Abelian groups of burden groups of burden 2. In the special case in which the structure has dp-rank 2, we show that the existence of an infinite definable…
Diophantine exponents are ones of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of…
Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…
We show that the Cantorvals connected with the geometric Cantor sets are not achievement sets of any series. However many of them are attractors of IFS consisting of affine functions.
This paper collects polynomial Diophantine equations that are simple to state but apparently difficult to solve.
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen…
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
In this paper we give a survey of what is currently known about Diophantine exponents of lattices and propose several problems.
Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…
The goal of the work is to take on and study one of the fundamental tasks studying Bidiophantine polygons (let us call a polygon Diophantine, if the distance between each two vertex of those is expressed by a natural number and we say that…
We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.
This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date…