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 Image may be NSFW.
Clik here to view. $\mathbb R^d$ with density at least Image may be NSFW.
Clik here to view. $\displaystyle (1-o(1))\frac{d \log d}{2^{d+1}}\,$ as Image may be NSFW.
Clik here to view. $d\to\infty$ This improves upon previous bounds for general Image may be NSFW.
Clik here to view. by a factor of order Image may be NSFW.
Clik here to view. $\log d$ 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 Image may be NSFW.
Clik here to view. $2^{-d}$ . (This is because when you double all spheres you will get a covering of space.) In 1905, this was improved by a factor of Image may be NSFW.
Clik here to view. 2 + o(1) by Minkowski. The first asymptotic improvement was by Rogers in 1947 and he gave a lower bound of Image may be NSFW.
Clik here to view. $\displaystyle 2^{-d} d c(1+o(1))$ . Roger’s value for the constant Image may be NSFW.
Clik here to view. was Image may be NSFW.
Clik here to view. c=2/e , and this value was improved to Image may be NSFW.
Clik here to view. 1.68 , Image may be NSFW.
Clik here to view., Image may be NSFW.
Clik here to view. 6/e (when Image may be NSFW.
Clik here to view. is divisible by 4), and then by Venkatesh to Image may be NSFW.
Clik here to view. c=65963 (when Image may be NSFW.
Clik here to view. is large; I wonder, how large?). Venkatesh also proved a lower bound of the form Image may be NSFW.
Clik here to view. $2^{-d} d \log \log d$ 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 Image may be NSFW.
Clik here to view. $2^{-(.599+o(1))d}$ .

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 Image may be NSFW.
Clik here to view., Image may be NSFW.
Clik here to view. $\Delta (G)$ denote the maximum degree of a vertex in Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. $\Delta_2 (G)$ denote the maximum codegree, namely, the maximum number of common neighbours a pair of distinct vertices in G can have. (Image may be NSFW.
Clik here to view. $\alpha(G)$ is the independence number of Image may be NSFW.
Clik here to view..)

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

Image may be NSFW.
Clik here to view. $\displaystyle \alpha (G) \ge (1-o(1))\frac {n \log \Delta}{\Delta},$

where Image may be NSFW.
Clik here to view. o(1) tends to 0 as Image may be NSFW.
Clik here to view. $\Delta \to \infty$ and we can take Image may be NSFW.
Clik here to view. $C=2^{-7}$ and Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. $\displaystyle 2^{-d} d c(1+o(1))$ lower bound using Shearer’s theorem.)

The graph G(X,r)

If Image may be NSFW.
Clik here to view. is a subset of Image may be NSFW.
Clik here to view. $\mathbb R^d$ the graph Image may be NSFW.
Clik here to view. G(X,r) has the set Image may be NSFW.
Clik here to view. as its set of vertices and two vertices are adjacent if their distance is at mosr Image may be NSFW.
Clik here to view.. The proof of the new sphere packing bound is based on discretization step, where you start with a huge ball Image may be NSFW.
Clik here to view. $\Omega$ and find a finite set of point Image may be NSFW.
Clik here to view. $X \subset \Omega$ such that Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. $(1+o(1)d \log d2^{-d}$ . 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 Image may be NSFW.
Clik here to view. $S^{d-1}$ of pairwise angle at least Image may be NSFW.
Clik here to view. $\theta$ . Denote by Image may be NSFW.
Clik here to view. $A(d,\theta)$ the maximal cardinality of such a spherical code. Let Image may be NSFW.
Clik here to view. $s_d(\theta)$ be the surface area of a spherical cup of radius Image may be NSFW.
Clik here to view. $\theta$ normalized by the surface area of the entire sphere Image may be NSFW.
Clik here to view. $S^{d-1}$ . (We assume that Image may be NSFW.
Clik here to view. $\theta < \pi/2$ .) Only recently, Jenssen, Joos, and Perkins improved the easy bound of Image may be NSFW.
Clik here to view. $1/s_d(\theta)$ (just look at a saturated family of spherical caps of radius Image may be NSFW.
Clik here to view. $\theta/2$ ) by a linear factor in the dimension. The new paper adds an additional Image may be NSFW.
Clik here to view. $\log d$ factor.