New proof of Poltoratskii gap theorem

The Beurling–Malliavin density appeared in 60th due to solution of completeness problem for the system of exponentials
$$ E(\Lambda):=\{e^{i\lambda t}\}_{\lambda\in\Lambda} $$
in $L^2(-\pi,\pi)$. It turns out that Beurling–Malliavin density is equal to radius of completeness of $E(\Lambda)$. Nevertheless, the lower Beurling-Malliavin density appeared in the literature only few years ago. M. Mitkovskii and A. Poltoratskii have proved that lower Beurling–Malliavin density corresponds to the so-called gap characteristic of $\Lambda$ (supremum of size of gaps of the Fourier transforms of measures supported on $\Lambda$) in the case when $\Lambda$ is separated. Proof by M. Mitkovskii and A. Poltoratskii based on theory of Toeplitz operators. We are able to find that there exists a direct way to show that this result is equivalent to the famous Beurling–Malliavin theorem for separated sequences. This gives us a new proof of gap theorem.
The report is based on joint work with A. Baranov and A. Ulanovskii.