Related papers: On the Multiple Packing Densities of Triangles

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has…

We consider the Dean-Kawasaki (DK) equation of overdamped Brownian particles that forms the basis of the stochastic density functional theory. Recently, the linearized DK equation has successfully reproduced the full Onsager theory of…

We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least ck^8 points of P, for some constant c, contains at…

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have…

Let $\omega=(-1+\sqrt{-3})/2$. For any lattice $P\subseteq \mathbb{Z}^n$, $\mathcal{P}=P+\omega P$ is a subgroup of $\mathcal{O}_K^n$, where $\mathcal{O}_K=\mathbb{Z}[\omega]\subseteq \mathbb{C}$. As $\mathbb{C}$ is naturally isomorphic to…

This paper presents new lower bounds for the lattice covering densities of simplices by studying the Degree-Diameter Problem for abelian Cayley digraphs. In particular, it proves that the density of any lattice covering of a tetrahedron is…

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight t^m to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s…

We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/2^n, which matches, up to a multiplicative constant, the best known density of a…

Let $V$ be a smooth, projective, rationally connected variety, defined over a number field $k$, and let $Z\subset V$ be a closed subset of codimension at least two. In this paper, for certain choices of $V$, we prove that the set of…

Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set…

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when…

For integers x and k, let T(x;2k) denote the number of twin prime pairs (p,p+2k) with a distance 2k<=2x**0.5 and p<=x (not p+2k<=x). Let Tg(x;2x**0.5) denote the average of T(x;2k) for all 2k<=2x**0.5. Logically, T(x;2k) should be a…

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent $\alpha > 0$, that characterizes the power-law scaling behavior of the structure factor $S(\mathbf{k})$ as a function of wavenumber $k\equiv|\mathbf{k}|$…

Sarnak's Density Conjecture is an explicit bound on the multiplicities of non-tempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue.…

From the Poisson-Dirichlet diffusions to the $Z$-measure diffusions, they all have explicit transition densities. In this paper, we will show that the transition densities of the $Z$-measure diffusions can also be expressed as a mixture of…

We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2),... n(k),.... Extending and improving previous work of Nurmela…

We consider the problem of decomposing some $t$-uniform hypergraph $G$ into copies of another, say $H$, with nonnegative rational weights. For fixed $H$ on $k$ vertices, we show that this is always possible for all $G$ having sufficiently…

Associated with a smooth, $d$-closed $(1, 1)$-form $\alpha$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0,…

We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(\gamma_p + o(1))n}$ with $\gamma_p < - 1/p$. This is the first exponential improvement in high dimensions since…

The $k$-coverage problem is to find the minimum number of disks such that each point in a given plane is covered by at least $k$ disks. Under unit disk condition, when $k$=1, this problem has been solved by Kershner in 1939. However, when…