Using the restricted invertibility theorem, Bourgain and Tzafriri have shown the existence of Riesz sequences of complex exponentials with positive density. In this talk we shall present an improvement of this result which relies on the solution to the Kadison-Singer problem by Marcus, Spielman, and Srivastava.

We show that every subset $\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\{ e^{i\lambda x}\} _{\lambda \in \Lambda}$ such that $\Lambda\subset\mathbb{Z}$ is a set with gaps between consecutive elements bounded by ${\displaystyle \frac{C}{\left|\mathcal{S}\right|}}$.

We also present higher dimensional analogues of this result. We explain how such results follow from a strengthening of the Feichtinger conjecture, which takes the form of a selection theorem for Bessel sequences with norms bounded from below. This talk is based on a joint work with Itay Londner.