The Heil–Ramanathan–Topiwala conjecture on time-frequency shifts

Let $f\in L^2(\operatorname{\mathbb{R}})$ be nonzero, and let $\Lambda\subset\operatorname{\mathbb{R}}^2$ be a finite set. For $\lambda=(x,\omega)\in\operatorname{\mathbb{R}}^2$, write $\pi(\lambda)f=M_\omega T_xf$ for the time-frequency shift, where $T_xf(t)=f(t-x)$ and $M_\omega f(t)=e^{2\pi i\omega t}f(t)$. The Heil–Ramanathan–Topiwala conjecture. The collection of functions

$$\mathcal{G}(f,\Lambda)=\{\pi(\lambda)f\}_{\lambda\in\Lambda}$$

is linearly independent. This is a fundamental conjecture in time-frequency analysis. The paper proves it for widely spaced sets, but the assertion in full generality remains open.