Uniqueness and non-uniqueness pairs for the fractional Laplacian

We study uniqueness and non-uniqueness pairs for the fractional Laplacian in the discrete setting. More precisely, under mild regularity assumptions on a function $f$ satisfying $f=0$ on $\mathrm{\Lambda }$ and $(-\mathrm{\Delta }{)}^{s}f=0$ on $M$, where $\mathrm{\Lambda },M\subset {\mathbb{R}}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. The uniqueness proofs combine decay estimates forced by discrete zero sets with analytic continuation, while the non-uniqueness results are obtained from an interpolation theorem.