Distinct distances on a sphere

Extension theorems and a connection to the Erdős-Falconer distance problem over finite fields

The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide a new $L^2\to L^r$ extension estimate for paraboloids in dimensions $d=4k+3$ and $q\equiv 3\mod 4$, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using \textit{the first association…

Topology and chromatic number of random $ε$-distance graphs on spheres

Given $0<α\leqπ$, $ε>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$ is $ε$-close to $α$. This model generalizes distance graphs on spheres, and also random Borsuk graphs. Topological tools are known to give tight bounds for the chromatic number of Borsuk graphs. We now study the efficiency of one of these topological invariants, namely the connectivity of Lóvasz's neighborhood complex, to bound the chromatic number of this model of random gr…

Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres

This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Nelson-Hadwiger problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not contain the Moser spindle as a subgraph. The construction of 5-chromatic graphs embedded in a sphere at two values o…

Distance regularity in buildings and structure constants in Hecke algebras

In this paper we define generalised spheres in buildings using the simplicial structure and Weyl distance in the building, and we derive an explicit formula for the cardinality of these spheres. We prove a generalised notion of distance regularity in buildings, and develop a combinatorial formula for the cardinalities of intersections of generalised spheres. Motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of Hecke operators arising from our generalised distance regularity, and prove isomorphisms between these algebras a…

How many $n$-vertex triangulations does the $3$-sphere have?

It is known that the $3$-sphere has at most $2^{O(n^2 \log n)}$ combinatorially distinct triangulations with $n$ vertices. Here we construct at least $2^{Ω(n^2)}$ such triangulations.

The surface of a sufficiently large sphere has chromatic number at most 7

We present a method to assign, for any radius $r$ greater than about 12.44, one of seven colors to each point in $\mathbb{R}^3$ lying at distance $r$ from the origin, such that no two points at unit distance from each other are assigned the same color. The existence of such a construction contrasts with the recent demonstration that, for any positive value $\varepsilon$, if no two points assigned the same color lie at any distance in $[1,1+\varepsilon]$ (and with certain other restrictions that are also satisfied with our coloring), then eight colors are needed for any finite $r\ge18$, even th…

Distinct and repeated distances on a surface and incidences between points and spheres

In this paper we show that the number of distinct distances determined by a set of $n$ points on a constant-degree two-dimensional algebraic variety $V$ (i.e., a surface) in $\mathbb R^3$ is at least $Ω\left(n^{7/9}/{\rm polylog} \,n\right)$. This bound is significantly larger than the conjectured bound $Ω(n^{2/3})$ for general point sets in $\mathbb R^3$. We also show that the number of unit distances determined by $n$ points on a surface $V$, as above, is $O(n^{4/3})$, a bound that matches the best known planar bound, and is worst-case tight in 3-space. This is in sharp contrast with the bes…

Graphs with three distinct distance eigenvalues

In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family of trees with distance eigenvalue $-1$, and determine all $\{C_3,C_4\}$-free connected graphs with three distinct distance eigenvalues of which the smallest one is equal to $-3$, which partially answers a problem posed by Koolen, Hayat and Iqbal [Linear Algebra Appl. 505 (2016) 97--108]. Furthermore, we characterize all trees with three distinct distance eigenvalues.

Distinct distances on non-ruled surfaces and between circles

We improve the current best bound for distinct distances on non-ruled algebraic surfaces in ${\mathbb R}^3$. In particular, we show that $n$ points on such a surface span $Ω\left(n^{32/39-\varepsilon}\right)$ distinct distances, for any $\varepsilon>0$. Our proof adapts the proof of Székely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in ${\mathbb R}^3$. Consider point sets ${\mathcal P}_1$ and ${\mathcal P}_2$ of respective sizes $m$ and $n$, such that each …

A note on distinct distance subsets

It is shown that given a set of $N$ points in the plane or on the sphere, there is a subset of size $\gtrsim N^{1/3}/\log N$ with all pairwise distances between points distinct.

The spectral excess theorem for distance-regular graphs having distance-$d$ graph with fewer distinct eigenvalues

Let $Γ$ be a distance-regular graph with diameter $d$ and Kneser graph $K=Γ_d$, the distance-$d$ graph of $Γ$. We say that $Γ$ is partially antipodal when $K$ has fewer distinct eigenvalues than $Γ$. In particular, this is the case of antipodal distance-regular graphs ($K$ with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs ($K$ with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with $d$ distinct eigenvalues) in terms of the spectrum and the mean number of vertices at m…

On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $Ω(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $Ω(n^{.6091})$ distinct distances.

Vanishing ideals of binary Hamming spheres

We consider the simplified Algebraic Normal Form (sANF) of Boolean functions vanishing on Hamming spheres centred at zero and the associated sANF vector. We show that this vector is periodic, leading to an efficient computation of the sANF and to specific formulas for particular cases. Moreover, we explicitly provide a connection to the binary M{ö}bius transform of the elementary symmetric functions. We conclude by presenting a method based on polynomial evaluation to bound the minimum distance of binary nonlinear codes. The same method can be used to compute the minimum distance and the weigh…

Distance-regular graphs where the distance-$d$ graph has fewer distinct eigenvalues

Let the Kneser graph $K$ of a distance-regular graph $Γ$ be the graph on the same vertex set as $Γ$, where two vertices are adjacent when they have maximal distance in $Γ$. We study the situation where the Bose-Mesner algebra of $Γ$ is not generated by the adjacency matrix of $K$. In particular, we obtain strong results in the so-called `half antipodal' case.

Sets with few distinct distances do not have heavy lines

Let $P$ be a set of $n$ points in the plane that determines at most $n/5$ distinct distances. We show that no line can contain more than $O(n^{43/52}{\rm polylog}(n))$ points of $P$. We also show a similar result for rectangular distances, equivalent to distances in the Minkowski plane, where the distance between a pair of points is the area of the axis-parallel rectangle that they span.

Distinct Distances: Open Problems and Current Bounds

We survey the variants of Erdős' distinct distances problem and the current best bounds for each of those.

三维欧氏空间中的Erdős不同距离问题

The Erdős distinct distances problem in $\mathbb{R}^3$

该研究证明三维欧氏空间中N个点至少确定N^(2/3-o(1))个不同距离，推进了Erdős不同距离问题的相关研究

A New Lower Bound for the Distinct Distance Constant

The reciprocal sum of Zhang sequence is not equal to the Distinct Distance Constant. This note introduces a $B_2$-sequence with larger reciprocal sum, and provides a more precise estimation of the reciprocal sums of Mian-Chowla sequence and Zhang sequence.

Measurable Chromatic Number of Spheres

We examine the measurable chromatic number of distance colorings on the surface of 2-dimensional spheres of varying radii, showing in particular that similar arguments to those used to raise lower bounds in the plane work for all but a countable set of radii. Furthermore, we show that measurable chromatic number as a function of the radius, or more generally the curvature, is not monotonic.

On sets with few distinct distances

It is widely believed that point sets in the plane which determine few distinct distances must have some special structure. In particular, such sets are believed to be similar to a lattice. This note considers two different ways to quantify this idea. Firstly, improving on a result of Hanson (see arXiv:1607.03442), it is proven that if $P= A \times A$ with $A \subset \mathbb R$ and $P$ determines $O(|A|^2)$ distinct distances, then $|A-A|=O\left(|A|^{2-\frac{2}{11}}\right)$. This result gives further evidence that cartesian products which determine few distinct distances have some additive str…