On syndetic Riesz sequences

TL;DR

This paper leverages the solution to the Kadison-Singer problem to construct Riesz sequences of exponentials on subsets of the torus, with explicit bounds on the frequency gaps, including a deterministic construction for open sets.

Contribution

It introduces a method to generate Riesz sequences with bounded gaps for any positive measure subset of the torus, extending previous results with explicit bounds and deterministic constructions.

Findings

Existence of Riesz sequences with bounded gaps for positive measure sets.

Explicit bounds on the gaps between frequencies in the sequences.

Deterministic construction of such sequences for open sets using quasicrystals.

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 ${\displaystyle \frac{C}{\left|\mathcal{S}\right|}}$. In the case when $\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $\Lambda$ can be deterministically constructed.