A Proof of the HRT Conjecture for Widely Spaced Sets

Given $f \in C_0(\mathbb{R}^n)$ and $\Lambda \subset \mathbb{R}^{2n}$ a finite set we demonstrate the linear independence of the set of time-frequency translates $\mathcal{G}(f, \Lambda) = \{\pi(\lambda)f\}_{\lambda\in \Lambda}$ when the time coordinates of points in $\Lambda$ are far apart relative to the decay of $f.$ As a corollary, we prove that for any $f \in C_0(\mathbb{R}^n)$ and finite $\Lambda \subset \mathbb{R}^{2n}$ there exist infinitely many dilations $D_r$ such that $\mathcal{G}(D_rf, \Lambda)$ is linearly independent. Furthermore, we prove that $\mathcal{G}(f, \Lambda)$ is linearly independent for functions like $f(t) = \frac{cos(t)}{|t|}$ which have a singularity and are bounded away from any neighborhood of the singularity.