Sphere packing in dimensions 8 and 24

Remarks

Maryna Viazovska proved the case of dimension $8$ in 2016. Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Viazovska proved the case of dimension $24$ soon after. Both proofs use the linear programming bound of Cohn and Elkies. If $f \colon \R^n \to \R$ is a suitable integrable function with $f(0) = \hat f(0) > 0$, $f(x) \le 0$ for $\abs{x} \ge r$, and $\hat f \ge 0$, then every packing of balls of radius $r/2$ has density at most $\operatorname{vol}(B(0, r/2))$. Viazovska built such a function for $r = \sqrt 2$ in dimension $8$ from modular forms and quasimodular forms, and the second paper did the same in dimension $24$. The density values follow from the lattices. $E_8$ has covolume $1$ and minimal length $\sqrt 2$, so the balls have radius $\sqrt 2/2$. The Leech lattice has covolume $1$ and minimal length $2$, so the balls have radius $1$.

Going deeper

For a student

The theorem completes the list of dimensions $1$, $2$, $3$, $8$, $24$ in which the sphere packing problem is solved. The Cohn–Elkies bound is a Fourier analytic version of the Delsarte bound for codes. Numerical computations by Cohn and Elkies suggested that it is sharp in dimensions $2$, $8$ and $24$. The work was to find the exact auxiliary function in dimensions $8$ and $24$. Viazovska received the Fields Medal in 2022, partly for this work.