On the uniqueness sets in the Fock space

TL;DR

This paper extends the understanding of uniqueness sets in the Fock space by demonstrating their stability under broad classes of perturbations, beyond previously known lattice-based configurations.

Contribution

It generalizes the class of perturbations under which sets remain uniqueness sets in the Fock space, beyond the integer lattice and uniform perturbations.

Findings

Uniqueness sets are stable under general perturbations.

The result broadens the class of known uniqueness sets.

Provides new conditions for sets to be uniqueness sets in Fock space.

Abstract

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.