It is well-known that $\{e^{i n t}\}_{n\in\mathbb Z}$ is an orthonormal basis for $L^2(-\pi,\pi)$. A theorem by Kadec (Kadec $1/4$ theorem) studies the perturbed exponential system:
If $\{\lambda_n\}$ is a sequence of real numbers for which $$|\lambda_n-n|\leqq L<\frac{1}{4}, \ \ n=0, \pm 1, \pm 2, \dots$$ then $\{e^{i \lambda_n t}\}_{n\in\mathbb Z}$ satisfies the Paley-Wiener criterion and so forms a Riesz basis for $L^2(-\pi,\pi)$.
It implies that a set $\{e^{i\lambda_n t} \}_{n\in \mathbb Z}$ is a Riesz basis of $L^2(-\pi,\pi)$ whenever $l=\sup_{n\in\mathbb Z} |n - \lambda_n | < \frac{1}{4}$. But it does not exclude that there are sequences $\lambda_n$ which have distance from the integers greater than $1/4$ and such that $\{e^{i\lambda_n t} \}_{n\in \mathbb Z}$ is a Riesz basis for $L^2(-\pi,\pi)$.
A recent work
De Carli, L., 2019. Concerning exponential bases on multi-rectangles in $\mathbb R^d$, in: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J.(Eds.), Topics in Classical and Modern Analysis: In Memory of YingkangHu. Springer International Publishing. Applied and Numerical HarmonicAnalysis. Chapter 3. arXiv version here
shows (see, in particular, Corollary 5.3 therein) that $l \geq \frac{1}{4}$ but under Corollary 5.3 $\{e^{i\lambda_n t} \}_{n\in \mathbb Z}$ is a Riesz basis of $L^2\left(-\pi,\pi\right)$.
I wonder if anyone knows other works that contain this kind of result, i.e. $l \geq \frac{1}{4}$ but $\{e^{i\lambda_n t} \}_{n\in \mathbb Z}$ still a Riesz basis for $L^2\left(-\pi,\pi\right)$.
