  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.
