Extension and restriction principles for the HRT conjecture

The HRT (Heil-Ramanathan-Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on $\mathbb{R}$ is linearly independent. This longstanding conjecture remains largely open even in the case when the function is assumed to be smooth. Nonetheless, the conjecture has been proved for some special families of functions and/or special sets of points. The main contribution of this paper is an inductive approach to investigate the HRT conjecture based on the following question. Suppose that the HRT is true for a given set of ($N$) points and a given function. We characterize the set of all new points such that the conjecture remains true for the same function and the set of $N+1$ points obtained by adding one of these new points to the original set. To achieve this we introduce a real-valued function whose global maximizers describe when the HRT is true. To motivate this new approach we re-derive the HRT for sets of $3$ points. Subsequently, we establish new results for points in $(1,n)$ configurations, and for a family of symmetric $(2,3)$ configurations. Furthermore, we use these results and the refinements of other known ones to prove that the HRT holds for certain families of $4$ points. In particular, we show that the HRT holds for any set of $4$ points and any real-valued Schwartz function.