*Thomas Hales*

The sphere packing problem is the second Fields medal problem I have worked on. The first was the Fundamental Lemma, solved by Ngô. Those who solve hard problems that I tried unsuccessfully to solve earn my undying admiration.

Inspired by Viazovska’s recent Fields medal, I searched my basement for any calculations of historical value. In fact, back in 1991, Conway suggested to me that to solve the sphere packing problem, we should view flat space as a sphere of infinite radius. I saw potential in the idea and made a few calculations between May and August that year.

I hesitate to show unpolished work that was never prepared for publication, having a distaste of publishing partial results. Nevertheless, here is a scan of a few handwritten notes from June 15, 1991, because they have a direct bearing on the eventual solution of the sphere packing problem in eight dimensions. I have not mentioned these notes to anyone until now. These many years later, the notes are in a rather presentable form. They anticipate the later work by Cohn-Elkies (1999-2003) that established the framework for Viazovska’s work. They describe the key role of the Fourier-Bessel transform (known also as the Hankel transform and equivalent to the Fourier transform for radial functions). The condition of the non-negativity of the Fourier-Bessel transform is stated. They study properties of functions q — now called Viazovska’s magic functions — and made a first guess of a formula for them in eight and twenty-four dimensions.

In the notes, the “guess for E_{8} is q(x) = sin(π x^{2})^{2}/(π^{2} x^{4} (1-x)), but it doesn’t work.”

Adjusting by √2 to compensate for our different conventions, the incorrect guess of the magic function in dimension eight from 1991 has the right schematic form, as shown by the schematic diagram on page 8 in Cohn’s Fields medal laudatio for Viazovska. Of course, it was a long journey from Conway’s idea of a sphere of infinite radius to Viazovska’s exact formula for the magic functions.

Graph of the first guess q(x/√2). As with the true magic function, this function has zeros at √n, for n=2,4,…, is positive for x<√2 and is non-positive for x≥√2. Normalization is q(0)=1.

The idea of the notes is simple. Odlyzko and Sloane solved the kissing number problem in dimensions eight and twenty four by using positive definite spherical functions for a unit sphere. Rescaling the unit sphere to infinite radius (or equivalently, letting the radius of the spherical caps tend to zero), Odlyzko and Sloane translates into a bound on the density of sphere packings in Euclidean space.

An easy lemma shows that the bound on the density of sphere packings asserted in the scanned notes is the same as the bound given by the main result of Cohn-Elkies (Theorem 3.1). The derivation of the bound in the notes is incomplete.