On syndetic Riesz sequences

Abstract

Applying the solution to the Kadison-Singer problem, we show that every subset $\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials ${\left\{e^{i\lambda x}\right\}}_{\lambda \in \Lambda }$ such that $\Lambda \subset \mathbb{Z}$ is a set with gaps between consecutive elements bounded by $\frac{C}{\left|\mathcal{S}\right|}$. In the case when $\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $\Lambda$ can be deterministically constructed.

Minor typo corrections To appear in Israel Journal of Mathematics