Density criteria for Fourier uniqueness phenomena in $\mathbf{R}^d$

We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for a Schwartz function and its Fourier transform on $A$ (assuming those values are rapidly decreasing). More general results of the same nature hold for Fourier uniqueness pairs. This is an analog in all dimensions of the work of Kulikov, Nazarov, and Sodin in dimension $1$. Our methods are unrelated. As an application of our results, we produce Fourier uniqueness sets in higher dimensions which are optimally well-separated (up to constants). Our techniques also give the first purely analytic construction of discrete Fourier uniqueness pairs in higher dimensions. For a concrete example, consider \begin{align*} A = \{\delta |n|^{t-1} n : n \in \mathbf{Z}^d\} \qquad \text{and} \qquad B = \{\delta |n|^{u-1} n : n \in \mathbf{Z}^d\}, \end{align*} where $t,u,\delta > 0$ and $t+u = 1$. We show that when $\delta$ is sufficiently small, $(A,B)$ is a Fourier uniqueness pair, but when $\delta$ is sufficiently large, there is an infinite-dimensional space of Schwartz functions $f$ with $f|_A = \hat{f}|_B = 0$.