Related papers: Minimum packing density for sets of four integers
We construct a dense packing of regular tetrahedra, with packing density $D > >.7786157$.
Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The…
We show for the first time that collectively jammed disordered packings of three-dimensional monodisperse frictionless hard spheres can be produced and tuned using a novel numerical protocol with packing density $\phi$ as low as 0.6. This…
This is the first in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
Given a nonempty finite multiset $S$ of positive integers, we wish to find a partially ordered set $P$ of minimal cardinality such that the multiset of cardinalities of all maximal chains in $P$ equals $S$. This paper establishes upper and…
A nonnegative integer is called a fertility number if it is equal to the number of preimages of a permutation under West's stack-sorting map. We prove structural results concerning permutations, allowing us to deduce information about the…
The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures…
How many permutations are needed so that every infinite-coinfinite set of natural numbers with asymptotic density can be rearranged to no longer have the same density? We prove that the density number $\mathfrak{dd}$, which answers this…
For a graph, $G$, and a vertex $v \in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \in V(G)$, the vertex sets…
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $\delta=\frac{8}{5\pi}\approx 0.509$. This implies that any set of (not…
We use Razborov's flag algebra method to show a new asymptotic lower bound for the minimal density $m_4$ of monochromatic $K_4$'s in any 2-coloring of the edges of the complete graph $K_n$ on $n$ vertices. The hitherto best known lower…
Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree $d$ whose Galois group is $S_d$. Let $(a_n)$ be a linearly recuresive sequence of integers which has $P$ as its characteristic polynomial. We prove, under…
A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $\sigma$ in $\pi_n$ converges to $1/|\sigma|!$ as $n\to\infty$. We prove that $\pi_1,\pi_2,\dots$ is quasirandom if and…
We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is…
A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of…
We consider hard-disc mixtures with disc sizes within ratio $\sqrt{2}-1$, that is, the small disc exactly fits in the hole between four large discs. For each prescribed stoichiometry of large and small discs, the densest packings are…
In this paper we prove an asymptotic lower bound for the sphere packing density in dimensions divisible by four. This asymptotic lower bound improves on previous asymptotic bounds by a constant factor and improves not just lower bounds for…
This is the second in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering…
In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.