Uniqueness and non-uniqueness pairs for the fractional Laplacian

We establish sufficient conditions on discrete subsets of $\mathbb{R}^d$ for them to form a uniqueness or a non-uniqueness pair for the fractional Laplacian. Specifically, assuming that $f=0$ on $\Lambda$ and that $(-\Delta)^sf=0$ on $M$, where $\Lambda, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.