Almost sure recovery in quasi-periodic structures

We study random perturbations of quasi-periodic uniformly discrete sets in the $d$-dimensional euclidean space. By means of Diffraction Theory, we find conditions under which a quasi-periodic set $X$ can be almost surely recovered from its random perturbations. This extends the recent periodic case result of Yakir from "Recovery the lattice from its random perturbations" IMRN 2020 (arXiv:200201508).Comment: 24 pages. Key words: quasi-crystals, mathematical diffraction, stationarity. Comments to V3: Final version. To appear in Journal of Statistical Physic