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](https://arxiv.org/pdf/1512.02275.pdf)
> 
> 
> 


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 $L^2\left(-\pi,\pi\right)$.



Kadec $1/4$ theorem and some examples with $l=\sup_{n\in\mathbb Z} |n - \lambda_n | \geq \frac{1}{4}$
There is a complete characterization of sequences $\lambda\_n$ for which which 
$e^{i\lambda\_n t}$ is a Riesz basis:
G. Semmler,
[Complete interpolating sequences, discrete Muckenhoupt condition, and conformal mapping](http://www.acadsci.fi/mathematica/Vol35/Semmler.html), which permits to construct many examples.


