Worked solution: Viazovska's modular form solution to sphere packing in dimensions 8 and 24

Step 2 of 9: The Cohn–Elkies linear programming bound

In plain words

Think of trying to prove no arrangement of balls can be denser than a specific target, without checking every possible arrangement one by one. Cohn and Elkies found a clever shortcut: if you can invent one special 'test function' with a few sign properties, it automatically caps the density of every packing at once, turning a search over infinitely many geometric arrangements into the much more tractable problem of constructing a single well-behaved function.

f(x) \le 0 \;(|x| \ge r), \quad \hat f(t) \ge 0, \quad f(0) = \hat f(0) \implies \Delta \le \mathrm{vol}\!\left(\tfrac{r}{2}B\right)

Detailed analysis

Cohn and Elkies showed that if a radial Schwartz function $f: \mathbb{R}^d \to \mathbb{R}$ satisfies $f(x) \le 0$ for $|x| \ge r$ , its Fourier transform satisfies $\hat f(t) \ge 0$ everywhere, and $f(0) = \hat f(0) > 0$ , then the packing density of unit balls in $\mathbb{R}^d$ is at most the density of balls of radius $r/2$ ; equality is attained if the packing's difference vectors are exactly the zeros of $f$ and $\hat f$ respectively.

Terms in this step

Fourier transform: A transform $\hat f$ that rewrites a function $f$ in terms of waves of different frequencies; a function and its Fourier transform are tightly linked, so controlling the sign of both at once is a strong constraint.
admissible / Schwartz function: A smooth function that, together with its Fourier transform, decays fast enough at infinity for all the relevant sums and integrals in the Cohn–Elkies bound to make sense.

Knowledge used in this step

Convex and discrete geometry