Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames

We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L=A(Z+iZ{)}^d$ for an invertible complex-valued matrix. As an application we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann-Fock space in $C ^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.