# Kadec $1/4$ theorem and some examples with $l=\sup_{n\in\mathbb Z} |n - \lambda_n | \geq \frac{1}{4}$

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

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.