  It was known to von Neumann in the 1950's that the integer lattice
$\mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later
demonstrated by Seip and Wallst\'en that a sequence of points $\Gamma$ that is
uniformly close to the integer lattice is still a uniqueness set. We show in
this paper that the uniqueness sets for the Fock space are preserved under much
more general perturbations.
