Marcelo Campos, Matthew Jenssen, Marcus Michelen and, and Julian Sahasrabudhe: Striking new Lower Bounds for Sphere Packing in High Dimensions

A few days ago, a new striking paper appeared on the arXiv

A new lower bound for sphere packing

by Marcelo Campos, Matthew Jenssen, Marcus Michelen, and Julian Sahasrabudhe

Here is the abstract:

We show there exists a packing of identical spheres in $\mathbb R^d$ with density at least $\displaystyle (1-o(1))\frac{d \log d}{2^{d+1}}\,$ as $d\to\infty$ This improves upon previous bounds for general by a factor of order $\log d$ and is the first asymptotically growing improvement to Rogers’ bound from 1947.

Let me tell you a little about this amazing result. Congratulations, Marcelo, Matthew, Marcus, and Julian!

In unrelated news: Congratulations to József Balogh, Robert Morris, Wojciech Samotij, David Saxton and Andrew Thomason for being awarded the Steele Prize for seminal contribution to research.

I am thankful to Zachary Chase for telling me about this development.

A brief history

Earlier lower bounds: Every saturated sphere packing (namely when you cannot squeeze in an additional sphere) have density $2^{-d}$ . (This is because when you double all spheres you will get a covering of space.) In 1905, this was improved by a factor of 2 + o(1) by Minkowski. The first asymptotic improvement was by Rogers in 1947 and he gave a lower bound of $\displaystyle 2^{-d} d c(1+o(1))$ . Roger’s value for the constant was c=2/e , and this value was improved to 1.68 , , 6/e (when is divisible by 4), and then by Venkatesh to c=65963 (when is large; I wonder, how large?). Venkatesh also proved a lower bound of the form $2^{-d} d \log \log d$ along a sparse sequence of dimensions. Here is the link to Venkatesh’s striking paper from a decade ago. The best upper bound going back to Kabatjanskii and Levenstein is of the form $2^{-(.599+o(1))d}$ .

A theorem about graphs

One interesting aspect of the new paper is the use of combinatorial methods and of graph-theory. The following theorem is closely related to results of Ajtai-Komlos-Szemeredi and of Shearer. For a graph , $\Delta (G)$ denote the maximum degree of a vertex in and $\Delta_2 (G)$ denote the maximum codegree, namely, the maximum number of common neighbours a pair of distinct vertices in G can have. ( $\alpha(G)$ is the independence number of .)

Theorem: Let be a graph on vertices, such that $\Delta(G)\le \Delta$ and $\Delta_2(G)\le C \Delta (\log \Delta)^{-c}$ . Then

$\displaystyle \alpha (G) \ge (1-o(1))\frac {n \log \Delta}{\Delta},$

where o(1) tends to 0 as $\Delta \to \infty$ and we can take $C=2^{-7}$ and c=7 .

This theorem (of much independent interest) is closely related to a theorem of Shearer for triangle free graphs, and an earlier theorem of Ajtai-Komlos-Szemeredi on Ramsey numbers r(3,n) . The technique of proof is known as the Rodl nibble (or the semi-random method). Moving from Shearer-type theorems to results about sphere packing was pioneered twenty years ago by Krivelevich, Litsyn, and Vardy. (Their paper reproved a $\displaystyle 2^{-d} d c(1+o(1))$ lower bound using Shearer’s theorem.)

The graph G(X,r)

If is a subset of $\mathbb R^d$ the graph G(X,r) has the set as its set of vertices and two vertices are adjacent if their distance is at mosr . The proof of the new sphere packing bound is based on discretization step, where you start with a huge ball $\Omega$ and find a finite set of point $X \subset \Omega$ such that G(X,r) has appropriate degrees and codegrees. The discretization step allows applying the graph theoretic theorem directly and to achieve a sphere packing based on the large independent set, guaranteed by the theorem, that G(X,r) contains.

Connection to physics and the replica method

The paper mentions the notion of amorphous sphere packings in physics and a prediction of Parisi and Zamponi based on the “replica method” that the largest density of “amorphous sphere packings” is $(1+o(1)d \log d2^{-d}$ . This is twice the value achieved in the paper, and the authors even predict that their graph theoretic theorem (and hence their sphere packing bound) can be improved by a factor 2. Roughly speaking, amorphous sphere packings have no long-term correlations.

Spherical codes and kissing numbers.

The paper contains similar constructions for spherical codes, and, in particular, for kissing numbers, and it added to recent improvements on these questions. A spherical code is a collection of points on the unit sphere $S^{d-1}$ of pairwise angle at least $\theta$ . Denote by $A(d,\theta)$ the maximal cardinality of such a spherical code. Let $s_d(\theta)$ be the surface area of a spherical cup of radius $\theta$ normalized by the surface area of the entire sphere $S^{d-1}$ . (We assume that $\theta < \pi/2$ .) Only recently, Jenssen, Joos, and Perkins improved the easy bound of $1/s_d(\theta)$ (just look at a saturated family of spherical caps of radius $\theta/2$ ) by a linear factor in the dimension. The new paper adds an additional $\log d$ factor.