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

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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, which permits to construct many examples.

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  • $\begingroup$ Thank you Alexandre Eremenko. I found the paper really interesting... I didn't know it! The main result of this paper is Theorem 6 which characterizes complete interpolating sequences. I'm not an expert in this filed but it seems that Th. 6 allows to construct many examples of the type I described above. But it also seems that this contruction depends on the knowledge of the zero set of the entire exponential function F. Am I right? $\endgroup$ – Mark Oct 13 '19 at 20:38
  • $\begingroup$ Theorem 6 gives a necessary and sufficient condition in completely elementary terms, without mentioning any entire function. $\endgroup$ – Alexandre Eremenko Oct 14 '19 at 23:06

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