Related papers: Bounds on antipodal spherical designs with few ang…

For positive integers $n\geq k\geq t$, a collection $ \mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and…

Spherical $t$-design is a finite subset on sphere such that, for any polynomial of degree at most $t$, the average value of the integral on sphere can be replaced by the average value at the finite subset. It is well-known that an…

We study query time bounds for the fundamental problem of estimating the kernel mean $\frac1{|X|}\sum_{x\in X}\mathbf{k}(x,y)$ of a query $y$ in a finite dataset $X\subset\mathbb{R}^d$ up to a prescribed additive error $\varepsilon$. The…

A set $C$ of unit vectors in $\mathbb{R}^d$ is called an $L$-spherical code if $x \cdot y \in L$ for any distinct $x,y$ in $C$. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and…

Spherical $t$-designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most $t$ via equal-weight quadrature. This concept has recently been extended to spherical $t$-design curves by the use…

In a projective plane $\Pi_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\mathcal{S}$ is saturating (or dense) if any point of $\Pi_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$. Modifying an…

A set system $\mathcal{F}$ is $t$-\textit{intersecting}, if the size of the intersection of every pair of its elements has size at least $t$. A set system $\mathcal{F}$ is $k$-\textit{Sperner}, if it does not contain a chain of length…

We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by (Lemmens and Seidel, 1973); namely, we use linear…

For $ E\subset \mathbb{F}_q^d$, let $\Delta(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $…

A construction for sphere packings is introduced that is parallel to the ``anticode'' construction for codes. This provides a simple way to view Vardy's recent 20-dimensional sphere packing, and also produces packings in dimensions 22,…

A finite set of distinct 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 $s$. In this paper…

We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…

Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that…

For a set $L$ of positive integers, a set system $\mathcal{F} \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\mathcal{F}$ the skew distance $sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\}$…

A finite set of the Euclidean space is called an $s$-distance set provided the number of Euclidean distances in the set is $s$. Determining the largest possible $s$-distance set for the Euclidean space of a given dimension is challenging.…

The absolute upper bound on the number of equiangular lines that can be found in $\mathbf{R}^d$ is $d(d+1)/2$. Examples of sets of lines that saturate this bound are only known to exist in dimensions $d=2,3,7$ or $23$. By considering the…

We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the open set condition. The bound is valid for all visible parts and…

A finite collection of unit vectors $S \subset \mathbb{R}^n$ is called a spherical two-distance set if there are two numbers $a$ and $b$ such that the inner products of distinct vectors from $S$ are either $a$ or $b$. We prove that if $a\ne…

The optimal exponentials of the thickness in the geometry rigidity inequality of shells represent the geometry rigidity of the shells. We obtain that the lower bounds of the optimal exponentials are $4/3,$ $3/2,$ and $1,$ for the hyperbolic…

Let $n$ and $t$ be positive integers with $t<n$, and let $q$ be a prime power. A partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of ${\rm PG}(n-1,q)$ that are pairwise disjoint. Let $r\equiv n\pmod{t}$…