Any sphere packing in R^8 has density at most 1.000001 pi^4/384

Geometry and Topology 2 Minutes

You need to know: Space {\mathbb R}^n, notation \|.\| for norm in {\mathbb R}^n, origin 0=(0,0,\dots,0)\in{\mathbb R}^n, ball B_n(0,R) with centre in the origin and radius R, volume |B_n(0,R)| of this ball, notation |A| for the number of elements in finite set A, scalar product \langle x,t \rangle:=\sum_{i=1}^n x_it_i in {\mathbb R}^n,  set {\mathbb C} of complex numbers, i=\sqrt{-1}, Fourier transform \hat{f}(t)=\int\limits_{{\mathbb R}^n} f(x) e^{-2\pi i \langle x,t \rangle}dx of an integrable function f:{\mathbb R}^n \to {\mathbb C}.

Background: A set S of points in {\mathbb R}^n such that \|x-y\|\geq 2 for all distinct x,y \in S is called a sphere packing. The center density of a sphere packing S is \delta(S) := \limsup\limits_{R\to\infty} \frac{|S\cap B_n(0,R)|}{|B_n(0,R)|}. The upper density of S is \Delta(S) = \delta(S)|B_n(0,1)|.

A function f:{\mathbb R}^n \to {\mathbb R} is called admissible if there exist positive constants \epsilon, C_1, C_2 such that |f(x)|\leq \frac{C_1}{(1+\|x\|)^{n+\epsilon}} and |\hat{f}(x)|\leq \frac{C_2}{(1+\|x\|)^{n+\epsilon}} for all x\in{\mathbb R}^n.

The Theorem: On 1st October 2001, Henry Cohn and Noam Elkies submitted to the arxiv and Annals of Mathematics a paper in which they proved the following result. Suppose f:{\mathbb R}^n \to {\mathbb R} is an admissible function, is not identically zero, and such that (i) f(x)\leq 0 for \|x\|\geq 1, and (ii) \hat{f}(t) \geq 0 (which implies that \hat{f}(t) is real) for all t\in{\mathbb R}^n. Then the center density of any sphere packing in {\mathbb R}^n is bounded above by \frac{f(0)}{2^n\hat{f}(0)}.

Short context: Finding the densest possible sphere packing in {\mathbb R}^n is a fundamental problem in geometry, which, before 2001, was solved only in dimensions n\leq 3. The Theorem provides a general method for proving upper bounds for sphere packing densities. In the same paper, Cohn and Elkies used the Theorem to derive improved upper bounds in dimensions 4 \leq n \leq 36. In particular, for n=8, they derive bound 1.000001\Delta_8, where \Delta_8=\frac{\pi^4}{384} is the density of densest known packing. In later works (see here and here), the Theorem was used to fully solve the sphere packing problem in dimensions 8 and 24.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 3.4 of this book for an accessible description of the Theorem.

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