On Falconer’s distance set problem in the plane

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

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Interior of sums of planar sets and curves

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000580 ◽

2018 ◽

Vol 168 (1) ◽

pp. 119-148 ◽

Cited By ~ 2

Author(s):

KÁROLY SIMON ◽

KRYSTAL TAYLOR

Keyword(s):

Unit Circle ◽

Cantor Set ◽

Full Measure ◽

Empty Interior ◽

Large Sets ◽

Distance Set

AbstractRecently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.

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Distance Sets of Urysohn Metric Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2012-022-4 ◽

2013 ◽

Vol 65 (1) ◽

pp. 222-240 ◽

Cited By ~ 6

Author(s):

N.W. Sauer

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Finite Metric Space ◽

Distance Set ◽

Distance Sets ◽

Separable Metric Space

Abstract.A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

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The Politics of Intimacy

Tales of Hope, Tastes of Bitterness ◽

10.5790/hongkong/9789888528042.003.0004 ◽

2019 ◽

pp. 65-81

Author(s):

Miriam Driessen

Keyword(s):

Local Community ◽

Sexual Intimacy ◽

Night Time ◽

Moral Principles ◽

Managerial Authority ◽

Corporate Hierarchy ◽

The Social ◽

Distance Set ◽

Set Up

Anxieties about the loss of integrity – the quality of having strong moral principles and the state of being undivided – are compounded by the increased intimacy between Chinese foremen and female members of the local community. Sexual relations, in particular, threaten to annul the carefully maintained distance between ‘us’ and ‘them’. Instances of sexual intimacy prompt Chinese managers to define notions of race and racial difference, and to reestablish their reputation as morally upright. What happens at night-time is consequential for the daytime encounters on the construction site. Sexual intimacy filters into management–labor relations and challenges the social distance set up between Chinese management and Ethiopian rank-and-file workers, on which the managerial authority and racial disparities of the corporate hierarchy depend.

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An Upper Bound for the Cardinality of an s -Distance Set in Euclidean Space

COMBINATORICA ◽

10.1007/s00493-003-0032-1 ◽

2003 ◽

Vol 23 (4) ◽

pp. 535-557 ◽

Cited By ~ 2

Author(s):

Etsuko Bannai ◽

Kazuki Kawasaki ◽

Yusuke Nitamizu ◽

Teppei Sato

Keyword(s):

Euclidean Space ◽

Upper Bound ◽

Distance Set

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Binary Representations of Regular Graphs

International Journal of Combinatorics ◽

10.1155/2011/101928 ◽

2011 ◽

Vol 2011 ◽

pp. 1-15

Author(s):

Yury J. Ionin

Keyword(s):

Regular Graph ◽

Line Graph ◽

Strongly Regular Graph ◽

Regular Graphs ◽

Spherical Representation ◽

Least Eigenvalue ◽

Vertex Set ◽

Strongly Regular ◽

Distance Set ◽

Hamming Space

For any 2-distance set in the n-dimensional binary Hamming space , let be the graph with as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in . The binary spherical representation number of a graph , or bsr(), is the least n such that is isomorphic to , where is a 2-distance set lying on a sphere in . It is shown that if is a connected regular graph, then bsr, where b is the order of and m is the multiplicity of the least eigenvalue of , and the case of equality is characterized. In particular, if is a connected strongly regular graph, then bsr if and only if is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph.

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Bounds on Antipodal Spherical Designs with Few Angles

The Electronic Journal of Combinatorics ◽

10.37236/9891 ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Zhiqiang Xu ◽

Zili Xu ◽

Wei-Hsuan Yu

Keyword(s):

Unit Sphere ◽

Finite Subset ◽

Upper Bound ◽

Maximum Size ◽

Tight Frames ◽

Previous Estimate ◽

Spherical Designs ◽

Equiangular Tight Frames ◽

Special Cases ◽

Distance Set

A finite subset $X$ on the unit sphere $\mathbb{S}^d$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X, \mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set $X$ for small $s$. Motivated by the method developed by Nozaki and Suda, for each even integer $s\in[\frac{t+5}{2}, t+1]$ with $t\geq 3$, we improve the best known upper bound of Delsarte, Goethals and Seidel. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

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