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We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denserthan a certain critical threshold, then $A$ is a Fourier uniqueness set, whileif $A$ is sparser, then uniqueness fails and one can prescribe arbitrary valuesfor a Schwartz function and its Fourier transform on $A$ (assuming those valuesare rapidly decreasing). More general results of the same nature hold forFourier uniqueness pairs. This is an analog in all dimensions of the work ofKulikov, Nazarov, and Sodin in dimension $1$. Our methods are unrelated. As anapplication of our results, we produce Fourier uniqueness sets in higherdimensions which are optimally well-separated (up to constants). Our techniquesalso give the first purely analytic construction of discrete Fourier uniquenesspairs 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 sufficientlysmall, $(A,B)$ is a Fourier uniqueness pair, but when $\delta$ is sufficientlylarge, there is an infinite-dimensional space of Schwartz functions $f$ with$f|_A = \hat{f}|_B = 0$.

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