On the center of distances (original) (raw)
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arXiv (Cornell University), 2024
We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane has all but a very small number of points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any noncollinear integer distance set of size n and a strong upper bound on the size of any integer distance set in [−N, N ] 2 with no three points on a line and no four points on a circle.
An non-centered asymmetric Cantor-like Set
Cornell University - arXiv, 2022
The ternary Cantor set C, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article will delve into the richness and the peculiarities of C through exploration of several variants and generalizations, and will provide an example of a non-centered asymmetric Cantor-like set.
On the ideal convergence of subsequences and rearrangements of a real sequence
Applied Mathematics Letters, 2010
In this article, we investigate the ideal convergence of subsequences and rearrangements of a real sequence. We also study some properties of the concepts of I ϕ -continuity, the I ϕ -limit point and the I ϕ -cluster point, where I ϕ is the family of sets which have ϕ-density zero.
A de Bruijn - Erdos theorem and metric spaces
Discrete Mathematics & Theoretical Computer Science
Combinatorics De Bruijn and Erdos proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.
Discrete subsets of R2 and the associated distance sets
We prove that a well-distributed subset of R 2 can have a separated distance set only if the distance is induced by a polygon. Basic definitions Separated sets. We say that S ⊂ R d is separated if there exists c separation > 0 such that ||x − y|| ≥ c separation for every x, y ∈ S, x = y. Here and throughout the paper, ||x|| = x 2 1 + • • • + x 2 d is the standard Euclidean distance. Well-distributed sets. We say that S ⊂ R d is well-distributed if there exists a C density > 0 such that every cube of side-length C density contains at least one element of S. K-distance. Let K be a bounded convex set, symmetric with respect to the origin. Given x, y ∈ R d , define the K-distance, ||x − y|| K = inf{t : x − y ∈ tK}. K-distance sets. Let A ⊂ R 2. Define ∆ K (A) = {||x − y|| K : x, y ∈ A}, the K-distance set of A. Notation: A B, with respect to the parameter R, means that there exists a positive constant C ǫ such that A ≤ C ǫ R ǫ B for any ǫ > 0. Similarly, A B means that there exists a C > 0 such that A ≤ CB, and A ≈ B means that A B and B A.
An Exploration Of The Generalized Cantor Set
International Journal of Scientific & Technology Research, 2013
In this paper, we study the prototype of fractal of the classical Cantor middle-third set which consists of points along a line segment, and possesses a number of fascinating properties. We discuss the construction and the self-similarity of the Cantor set. We also generalized the construction of this set and find its fractal dimension.
On the arithmetic sum of middle-cantor sets
Proyecciones (Antofagasta), 1995
In this article we study the arithmetic su m { difference) set K a + K {3 in 1 = [0, 1] in terms of the parameters (a, {3) E I X I, where K a and K fi are middle-Cantor sets contained in 1. We describe two regions, A and B, in the parameter space (a, {3) where the characterízation of the arithmetíc su m set K a + K fi ís gíven.
Metric characterizations of some subsets of the real line
Matematičnì studìï, 2023
A metric space (X, d) is called a subline if every 3-element subset T of X can be written as T = {x, y, z} for some points x, y, z such that d(x, z) = d(x, y) + d(y, z). By a classical result of Menger, every subline of cardinality ̸ = 4 is isometric to a subspace of the real line. A subline (X, d) is called an n-subline for a natural number n if for every c ∈ X and positive real number r ∈ d[X 2 ], the sphere S(c; r) := {x ∈ X : d(x, c) = r} contains at least n points. We prove that every 2-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup G ⊆ R, a metric space (X, d) is isometric to G if and only if X is a 2-subline with d[X 2 ] = G + := G ∩ [0, ∞). A metric space (X, d) is called a ray if X is a 1-subline and X contains a point o ∈ X such that for every r ∈ d[X 2 ] the sphere S(o; r) is a singleton. We prove that for a subgroup G ⊆ Q, a metric space (X, d) is isometric to the ray G + if and only if X is a ray with d[X 2 ] = G +. A metric space X is isometric to the ray R + if and only if X is a complete ray such that Q + ⊆ d[X 2 ]. On the other hand, the real line contains a dense ray X ⊆ R such that d[X 2 ] = R + .
Excursions on Cantor-Like Sets
The ternary Cantor set C, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article we will explore the richness, the peculiarities and the generalities that has C and explore some variants and generalizations of it. For a more systematic treatment the Cantor like sets we refer to our previos paper [7].