Some Results in Spectral Synthesis Over ZNᵈ

A classical result due to Agranovsky and Narayanan ({AN04}) says that if the support of the Fourier transform of f: Rⁿ → C is carried by a smooth measure on a d-dimensional manifold M, and f ∈ Lᵖ( Rᵈ) for p ≤ 2n/d, then f is identically equal to $0$. In this paper, we investigate an analogous problem for functions f: ZNᵈ → C. Bourgain's celebrated result on Λₚ sets ({Bou89}), random constructions ({Bab89}), and connections with the theory of exact signal recovery ({DS89}, {MS73}, {IKLM24}, {IM24}) play an important role.