The sphere packing problem in dimension $8$

Abstract

In this paper we prove that no packing of unit balls in Euclidean space R 8 has density greater than that of the E 8 -lattice packing.We define the density of a packing P as the limit superior ∆ P := lim sup r→∞ ∆ P (r).The number be want to know is the supremum over all possible packing densities ∆ d := sup P⊂R d sphere packing ∆ P , 1 arXiv:1603.04246v2 [math.NT]

A note on Schwartz functions and modular forms

“…Example. For d = 8 we can let n = 1 and find that φ(z) = E 2 2 E 2 4 − 2E 2 E 4 E 6 + E 2 6 which matches the function found in [11].…”

“…The Schwartz functions constructed by Viazovska, Cohn, Kumar, Miller, Radchenko, and Viazovska, and Cohn and Gonçalves in [11], [5], [6], and [3] showed that there is a surprising and beautiful connection between modular forms and various problems related to sphere packing. It is natural to seek a better understanding of these functions, and to aim for a general framework.…”

$E_8$, the most exceptional group

“…It has been known for a long time that it is the lattice with the densest sphere packing in R 8 , and the sphere packing with the highest kissing number [38]. Recently, Viazovska showed that it gives the densest sphere packing in R 8 , among all possible packings both lattice and nonlattice [163]. At the time of her work, the densest packing was unknown in R n for all n ≥ 4!…”

Character tables and the problem of existence of finite projective planes

“…The Delsarte LP‐bound has often proved fruitful when dealing with such problems, e.g. in the context of sphere‐packing or in the maximum number of code‐words in error correcting codes . It is based on the observation that the function is positive definite on the group G .…”