# From sphere packing to Fourier interpolation

## Abstract

Viazovska’s solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska.

## 1. Sphere packing

The sphere packing problem asks how densely congruent spheres can be packed in Euclidean space. In other words, what fraction of space can be filled with congruent balls if their interiors are required to be disjoint?Footnote^{1} Everyone can pack spheres intuitively in low dimensions: the optimal two-dimensional packing is a hexagonal arrangement, and optimal three-dimensional packings are stacks of optimal two-dimensional layers, nestled together as closely as possible into the gaps in the layers (see Figure 1.1).

^{1}

To make this question precise, we could take the limit as of the density for packing unit spheres in a sphere of radius or a cube of side length , We would obtain the same limit for any reasonable container (see, for example, .Reference 7).

In fact, these packings are known to be optimal. The two-dimensional problem was solved by Thue Reference 42Reference 43, with a more modern proof by Fejes Tóth Reference 21, and the three-dimensional problem was solved by Hales Reference 23. The two-dimensional proof is not so complicated, but the three-dimensional proof is difficult to check, because it relies on both enormous machine calculations and lengthy human arguments in a sequence of papers. To give a definitive demonstration of its correctness, Hales and a team of collaborators have produced a formally verified proof Reference 24, i.e., a proof that has been algorithmically verified using formal logic.

On the one hand, the solution of the three-dimensional sphere packing problem is a triumph of modern mathematics, a demonstration of humanity’s ability to overcome even tremendously challenging obstacles. On the other hand, to a general audience it can sound like a parody of pure mathematics, in which mathematicians devote immense efforts to proving an intuitively obvious assertion. It’s natural to feel discouraged about the future of a subfield in which it’s easy to guess the answer and almost impossible to prove it. For comparison, a rigorous solution of the four-dimensional sphere packing problem remains far out of reach. If the difficulty increases as much from three to four dimensions as it did from two to three, then humanity may never see a proof.

One noteworthy change as we move to higher dimensions is that we lose much of our intuition, and the answer is no longer easy to guess. For example, it is not always true that we can obtain an optimal packing in by stacking optimal layers (see -dimensionalReference 16 for details). In sufficiently high dimensions, there are no conjectures for optimal packings, the best upper and lower bounds known for the packing density differ by an exponential factor in the dimension, and we cannot even predict whether the densest packings should be crystalline or disordered. In short, we know shockingly little about how spherical particles behave in high dimensions. Of course this means there are plenty of intriguing phenomena to explore.

Certain dimensions stand out in the midst of this ignorance as having exceptionally dense packings. The most amazing of all are eight and twenty-four dimensions, which feature the root lattice and the Leech lattice (We will not construct these lattices here; see .Reference 17Reference 20Reference 41 for constructions.) Recall that a lattice in is just a discrete subgroup of rank in other words, for each basis ; …, , of the set ,

is a lattice. Every lattice leads to a sphere packing by centering congruent spheres at the lattice points, with the radius chosen as large as possible without overlap. Lattice packings are common in low dimensions, but there is no reason to expect an optimal packing to have this sort of algebraic structure in general. For example, in the best packing known, the aptly named Best packing Reference 5, has density more than 8% greater than any known lattice packing in By contrast, the . and Leech lattices yield impressively dense packings with extraordinary symmetry groups, and their density and symmetry are so far out of the ordinary that it is difficult to imagine how they could be improved.

In 2016 Maryna Viazovska Reference 44 solved the sphere packing problem in with an innovative use of modular forms, which was soon extended to as well Reference 13; both and the Leech lattice do indeed turn out to be optimal sphere packings. These are the only cases in which the sphere packing problem has been solved above three dimensions. Although the proofs require more machinery than those in two or three dimensions, most notably the theory of modular forms, they are much shorter and simpler than one might fear. Viazovska’s proof dispelled the gloomy possibility that higher-dimensional sphere packing could be beyond human understanding, and she was awarded a Fields Medal in 2022 for this line of work.

In addition to her breakthrough in sphere packing, Viazovska’s modular form techniques have led to unexpected consequences, such as interpolation theorems showing that a radial function can be reconstructed from the values of and its Fourier transform on certain discrete sets of points. Although Fourier interpolation may sound rather far afield from sphere packing, it turns out to be closely connected. In this article, we’ll explore how Viazovska’s work led to this connection and how to prove a fundamental interpolation theorem of Radchenko and Viazovska Reference 38. For comparison, Reference 32, Reference 9, Reference 45, Reference 46, and Reference 10 are expositions of her work that focus on other themes.

## 2. From sphere packing to Fourier analysis

The connection between packing problems and Fourier analysis originated in the work of Delsarte Reference 19 on linear programming bounds for error-correcting codes. For sphere packings in Euclidean space, a continuous analogue of Delsarte’s work was developed by Cohn and Elkies Reference 11. The quality of this bound depends on the choice of an auxiliary function satisfying certain inequalities, and Viazovska’s breakthrough amounted to figuring out how to optimize that choice.

We will normalize the Fourier transform of an integrable function by

where denotes the usual inner product on We’ll generally restrict our attention to Schwartz functions, i.e., infinitely differentiable functions . such that for all real numbers and nonnegative integers …, , ,

as

The linear programming bound is the following method for producing a density bound from a suitable auxiliary function

It is far from obvious how to produce good auxiliary functions

In most dimensions, the linear programming bound seems nowhere near sharp, but the upper and lower bounds appear to touch in eight and twenty-four dimensions. Cohn and Elkies conjectured that they were equal in those cases, and the solutions of the sphere packing problem in these dimensions come from proving this conjecture.Footnote^{2} For comparison, it is known that the linear programming bound cannot be sharp in dimensions three through five Reference 33, six Reference 18, twelve, or sixteen Reference 15, and it is likely that the only sharp cases are dimensions one, two, eight, and twenty-four.

^{2}

The linear programming bound also seems to be sharp in two dimensions, but no proof is known, despite the fact that the two-dimensional sphere packing problem itself can be solved by elementary means.

The optimal auxiliary functions in eight and twenty-four dimensions have come to be known as magic functions, because obtaining an exact solution in these dimensions feels like a miracle. To see how this miracle comes about, we will examine a proof of Theorem 2.1 for the special case of lattice packings. It is based on the Poisson summation formula, which states that

for every Schwartz function

The proof for more general packings is similar in spirit, but it applies Poisson summation to periodic packings given by unions of translates of a lattice. See Reference 11 or Reference 9 for the details.

Note that the proof of Theorem 2.1 does not actually require

What sort of function

It turns out that the

Now the whole problem comes down to constructing magic functions with these roots. That might not seem so difficult, but controlling the behavior of

Viazovska gave a remarkable construction of the eight-dimensional magic function in terms of modular forms, which are a class of special functions defined on the upper half-plane

which converges because

The first identity follows immediately from the defining series, while the second is more subtle and will be proved below. In this equation, we have to choose the branch for

To prove that

with

which is the complex generalization of the fact that the Fourier transform of a wide Gaussian is a narrow Gaussian, and vice versa. Now Poisson summation says that

because

and thus

The functions

Together with analyticity and some growth conditions, the identities Equation 2.1 say that *modular form of weight * for the group

Instead of examining the details of Viazovska’s construction, let’s think about a bigger picture. We know the eight-dimensional magic function

as in Figure 2.2. Viazovska conjectured that this data, together with the nonzero value

In particular, up to scaling the magic function is the interpolation basis function

Theorem 2.2 is similar in spirit to other interpolation theorems in mathematics. The simplest and most famous of these theorems is Lagrange interpolation, which says that a polynomial in one variable of degree less than

so that every polynomial

Lagrange interpolation can be generalized to Hermite interpolation, which takes into account derivatives along similar lines to Theorem 2.2: a polynomial

One important relative of Lagrange interpolation is Shannon sampling, which in the case of Schwartz functions

This theorem plays a crucial role in information theory, since it says that a band-limited signal (i.e., one with a limited range of frequencies) is determined by periodic samples. It’s worth noting that the product formula

is analogous to the products Equation 2.3 in the Lagrange interpolation basis. Much is known about Shannon sampling and its variations; see, for example, Reference 29 and the references cited therein.

Both Lagrange interpolation and Shannon sampling rely on a notion of size. We measure the size of a polynomial by its degree, and the size of a band-limited function by its bandwidth, the smallest ^{3}

^{3}

Furthermore, size is related to growth at infinity. For degrees of polynomials this is clear, while a band-limited function of bandwidth

Puzzlingly, Theorem 2.2 shows no sign of a similar notion of size. It is reminiscent of Shannon sampling, in that it takes into account both

## 3. First-order Fourier interpolation

How does one prove an interpolation theorem like Theorem 2.2? We’ll examine a technically simpler variant due to Radchenko and Viazovska, which is important in its own right and a beautiful illustration of Fourier interpolation. It deals with functions of one variable (so “radial” becomes “even”), and it studies interpolation to first order, without derivatives. This first-order interpolation theorem does not seem to have any applications to sphere packing, but it’s a fundamental fact about Fourier analysis, and it is remarkable that it was not known until well into the 21st century.

There is also a corresponding theorem about odd functions Reference 38, Theorem 7, which can be proved in almost the same way. We’ll focus on even functions here for simplicity. Note also that the root spacing has changed from

As a consequence of this interpolation theorem, if an even Schwartz function

In the remainder of this section, we’ll sketch a proof of Theorem 3.1. The sketch will omit a number of analytic details, but it will outline the techniques and explain where additional work is required.

The central question is where the interpolation basis

In particular,

It turns out that this formula is now irredundant, with no additional linear relations between the values

These constraints let us get a handle on

We’ll use the same semicircular contour of integration in all integrals from

and satisfies the functional equations