Asymptotically orthonormal basis of complex exponentials

A sequence of vectors $\{x_k\}$ in a Hilbert space is called an asymptotically orthonormal sequence (AOS) if there is a $N \in \mathbb N$ such that for all $n \geq N$, there are constants $a_n, b_n >0$ such that $$a_n \displaystyle\sum_{k\geq n} |c_k|^2 \leq \left\|\sum_{k\geq n} c_k x_k \right\|^2 \leq b_n \displaystyle\sum_{k\geq n} |c_k|^2, $$ for every $\{c_k\} \in \ell^2$ and $\displaystyle\lim_{n\rightarrow \infty} a_n = 1$, $\displaystyle\lim_{n\rightarrow \infty} b_n = 1$. If $N=1$, then the sequence is called an asymptotically orthonormal basis (AOB). Given a sequence of points $\{\lambda_n\} \subset \mathbb R$, when can we say that the corresponding family of exponentials $\{e^{i\lambda_nt}\}$ forms an AOB in $L^2[0,1]$? Recent developments by Mitkovski, using Toeplitz operators to prove certain basis properties for the family $\{e^{i\lambda_nt}\}$ give a new direction to this area of research. We will discuss some latest results based on this approach. This is joint work with Emmanuel Fricain.