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 $