The Fourier restriction and Kakeya problems over rings of integers modulo $N$

The Fourier restriction and Kakeya problems over rings of integers modulo $N$, Discrete Analysis 2018:11, 18 pp.

The Fourier restriction problem is the following general question. Suppose one has a smooth compact hypersurface $S$ in $\mathbb R^d$. Then the surface measure on $S$ allows one to define the inverse Fourier transform for functions from $S$ to $\mathbb C$, and the problem is to determine for which pairs $(p,q)$ it is bounded as a function from $L_p(S)$ to $L_q(\mathbb R^d)$.

The problem arises naturally for many reasons. As a simple example, consider the Helmholtz equation $\nabla^2A-kA=0$, where $A$ is a function defined on $\mathbb R^3$. Since the Fourier transform of $\frac{\partial f}{\partial x}$ is $ix\hat f$, if we take the Fourier transform, then this equation becomes
$$(k-x^2-y^2-z^2)\hat A=0,$$
the general solution of which is any function $\hat A$ that is zero outside the sphere $x^2+y^2+z^2=k$. This tells us in turn that the general solution of the original equation will be the inverse Fourier transform of any (suitably nice) function that is defined on the sphere, so if we want to understand solutions of the Helmholtz equation, then we need to understand the behaviour of the inverse Fourier transform in this situation.

The Kakeya problem asks how small (in various senses) a set can be if for every direction it contains a line in that direction: such sets are called Kakeya sets. To see (non-rigorously) the connection with the restriction problem, suppose that we have a smooth function defined on the sphere. If we zero in on a very small portion of the function, we will have a function that is approximately linear on a small disc. The inverse Fourier transform of the characteristic function of that small disc will be a line normal to the disc, and the inverse Fourier transform of the linear function will be an atomic measure, so by the convolution law, the inverse Fourier transform of the linear function restricted to the disc will be a function defined on some translate of the line normal to the disc. Since the sphere has normal vectors in every direction, we have thus ended up with a Kakeya set.

There are many specific questions of great interest connected with the restriction and Kakeya problems. One of the most famous is whether the Hausdorff dimension of a Kakeya set in $\mathbb R^d$ must be $d$. This is known to be the case when $d=2$ but not otherwise. (The current record when $d=3$, due to Nets Katz and Joshua Zahl, is $5/2+\epsilon_0$ for some small positive constant $\epsilon_0$ [2].)

A natural impulse when studying questions like these is to find discrete versions of the questions that one hopes will capture the essential difficulties of the problem but avoid certain technical ones. In the case of the Kakeya problem, for instance, the problem can be usefully reformulated as one about the volume of a union of long thin cylinders, but one then runs into the difficulty that if two cylinders are pointing in almost the same direction, then they can have a much larger overlap than if there is a significant angle between them, and this has to be taken into account in calculations. One attempt to get rid of this difficulty was to ask a version of the problem for $\mathbb F_p^d$ instead, where two distinct lines always intersect in at most one point. The corresponding problem was solved in a remarkable and highly influential paper of Dvir, using the polynomial method [1]. However, it quickly became clear that the differences between the finite-field problem and the Euclidean one were more than merely technical, so Dvir’s breakthrough has not led to a solution of the latter.

The purpose of this paper is to consider a different discretization of the problem that might have a better chance of leading to a solution of the Euclidean problem. This time the setting for the problem is $(\mathbb Z/N\mathbb Z)^d$, where $N$, instead of being prime, is highly composite. When that is the case, the cyclic group $\mathbb Z/N\mathbb Z$ has many subgroups, and we regard elements $x_1,\dots,x_k$ as being “close” if there is a coset of a small subgroup that contains all of them. (In particular, two points $x$ and $y$ are close if the highest common factor of $x-y$ and $N$ is large.)

To justify this choice of setting the authors show (i) that the Euclidean questions have natural analogues for $(\mathbb Z/N\mathbb Z)^d$, (ii) that many of the Euclidean questions become simpler and cleaner for these analogues, and most importantly (iii) that the $(\mathbb Z/N\mathbb Z)^d$ appears to resemble the Euclidean case more closely than the $\mathbb F_p^d$ case. To back up this third assertion, they show that some of the known results concerning the Euclidean restriction problem carry over to the $(\mathbb Z/N\mathbb Z)^d$ version, and many of the arguments can also be closely mirrored. Even when this is not the case, the questions that arise when studying the $(\mathbb Z/N\mathbb Z)^d$ version are interesting in their own right. Only time will tell whether or not the $(\mathbb Z/N\mathbb Z)^d$ model will be of genuine help with the Euclidean problems, but the authors show that it is an approach that is well worth investigating.

[1] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), 1093-1097, or arXiv 0803.2336

[2] N. Katz and J. Zahl, An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb R^3$, arXiv 1704.07210