Zero set of Zak transform and the HRT Conjecture

Abstract: The HRT (Heil-Ramanathan-Topiwala) posits the linear independence of any set of nonzero square-integrable vectors obtained from a single nonzero vector $f$ by applying a finite set of time-frequency shift operators. In this short note, we present findings centered on the zero set of the Zak transform of $f$, and a distinct arrangement involving a finite set of $N$ points on an integer lattice in the time-frequency plane, excluding a specific point.