Multidimensional Fourier Quasicrystals I. Sufficient Conditions

We derive sufficient conditions for an atomic measure ${\sum }_{\lambda \in \mathrm{\Lambda }}{m}_{\lambda }{\delta }_{\lambda },$ where $\mathrm{\Lambda }\subset {\mathbb{R}}^n,$ $m_{\lambda }$ are positive integers, and ${\delta }_{\lambda }$ is the point measure at $\lambda ,$ to be a Fourier quasicrystal, and suggest why they may also be necessary. These conditions extend the necessary and sufficient conditions derived by Lev, Olevskii, and Ulanovskii for $n=1.$ Our methods exploit the toric geometry relation between Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii.