Densest packings of translates of strings and layers of balls

TL;DR

This paper establishes new upper bounds for the density of packings of translates of certain string and layer configurations of unit balls in 3D and 4D, proving the first non-lattice packing density bounds in four dimensions.

Contribution

It introduces novel density bounds for packings of translates of specific string and layer structures of unit balls in 3D and 4D, including the first proof of a conjectured bound in 4D.

Findings

Maximum density for 3D packings with string configurations

Maximum density for 4D packings with layered configurations

Application of $(r,R)$-system theorem to packing problems

Abstract

Let $L \subset {\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \in [2, 2\sqrt{2}]$. Then a packing of unit balls in ${\Bbb R}^3$ consisting of translates of $L$ has a density at most $\pi /(3d\sqrt{3-d^2})$, with equality for a certain lattice packing of unit balls. Let $L \subset {\Bbb R}^4$ be the union of unit balls, whose centres lie on the $x_3x_4$ coordinate plane, and form either a square lattice or a regular triangular lattice, of edge length $2$. Then a packing of unit balls in ${\Bbb R}^4$ consisting of translates of $L$ has a density at most $\pi ^2/16$, with equality for the densest lattice packing of unit balls in ${\Bbb R}^4$. This is the first class of non-lattice packings of unit balls in ${\Bbb R}^4$, for which this conjectured upper bound for the packing density of balls is proved. Our main tool for the proof is a theorem on $(r,R)$-systems in ${\Bbb R}^2$. If $R/r \le 2 \sqrt{2}$, then the Delone triangulation associated to this $(r,R)$-system has the following property. The average area of a Delone triangle is at least $\min \{ V_0, 2r^2 \} $, where $V_0$ is the infimum of the areas of the non-obtuse Delone triangles. This general theorem has applications also in other problems about packings: namely for $2r^2 \ge V_0$ it is sufficient to deal only with the non-obtuse Delone triangles, which is in general a much easier task. Still we give a proof of an unpublished theorem of L. Fejes T\'oth and E (=J.) Sz\'ekely: for the $2$-dimensional analogue of our question about equidistant strings of unit balls, we determine the densest packing of translates of an equidistant string of unit circles with distance $2d$, for the first non-trivial interval $2d \in (2{\sqrt{3}},4)$.