We study random perturbations of quasi-periodic uniformly discrete sets in the d-dimensional euclidean space. By means Diffraction Theory, we find conditions under which a set X can be almost surely recovered from its perturbations. This extends recent periodic case result Yakir (Int Math Res Notices 1–19, 2020).

Two counterexamples, addressing questions raised in Adamczak (2019) and Poly Zheng (2019), are provided. Both counterexamples related to chaoses. Let F n = Y + Z , where the random variables belong different chaoses of uniformly bounded degree. It may be that ? a . s 0 L 2 ? E { sup | } < ? for some > yet fails converge a.s.