Questions about various types of basis properties of the sequence of complex exponentials $\{e^{i\lan t}\}$ have a very long history, with origins in the work of Paley, Wiener, and Levinson. In this talk I will present a characterization of the basis properties of this sequence in terms of the invertibility properties of a certain naturally associated Toeplitz operator. Using this characterization I will describe the radius of $l^2$-dependence of $\{e^{i\lan t}\}$, i.e., the supremum of all $c>0$ for which $\{e^{i\lan t}\}$ is $l^2$-dependent in $L^2[0,c]$. Namely, I'll show that this radius is equal to the interior Beurling-Malliavin density of the frequency sequence $\{\lan\}_{n\in\Z}$. I will also show how this Toeplitz operator method can be used to solve other classical problems in function theory.