On geometry of the unit ball of Paley-Wiener space over two intervals

Given a Banach space $X$, denote by $b(X)$ its unit ball, $b(X):=\{f\in X, \|f\|\leq 1\}$. An element $f\in X$ is called an extreme point of $b(X)$, if it is not a proper convex combination of two distinct points of $b(X)$. An element $f$ in $b(X)$ is an exposed point of $b(X)$, if there exists a functional $\phi\in X^\ast$ such that $\|\phi\|= 1$ and the set $\{g\in X : \phi(g)=1\}$ consists of one element, $f$. There is a large number of papers studying extreme and exposed points in different function spaces. In particular, the classical theorem of K. de Leeuw and W. Rudin (1958) states that the extreme points of the unit ball of the Hardy space $H^1$ on the unit disk are precisely the outer functions $f\in H^1$ with $\|f\|_1=1$. On the other hand, no description of the exposed points of ball$(H^1)$ is known.
Let $S$ be a compact set on the real line. Denote by $PW_S^1$ the space of integrable functions on the real line whose Fourier transform vanishes outside $S$ equipped with the $L^1$-norm. We are interested in the following
Problem (K. Dyakonov, 2021). Describe the sets of extreme and exposed points of $b(PW_S^1)$.
When $S=[-\sigma,\sigma],\sigma>0,$ is a single interval, a complete solution of this problem was obtained by K. Dyakonov in 2000. In particular, a function $f$ is an extreme point of $b(PW_S^1)$ if and only if $\|f\|_1=1$, at least one of the points $\sigma,-\sigma$ lies in the (closed) spectrum of $f$ and $f$ has no pairs of zeros symmetric with respect to the real line.
We consider the spectra $S$ which consist of two symmetric intervals,
$$ S:=[-\sigma,-\rho]\cup[\rho,\sigma]=[-\sigma,\sigma]\setminus(-\rho,\rho),\quad 0<\rho<\sigma. $$
We say that $S$ has the gap $(-\rho,\rho)$.
It turns out that the structure of the set of extreme and the set of exposed points of $b(PW_S^1)$ depends on the size of gap. If $\rho>\sigma/2$ (‘long gap’), then the description of these sets is somewhat similar to the one given by K. Dyakonov for the case of single interval. However, if $0<\rho<\sigma/2$ (‘short gap’), then the structure of these sets becomes more complicated. For the case of long gap, an essential step of the proof is to show that the exponential system with frequencies at the symmetric zeros of $f$ is not complete in the space $L^2$ on some proper subinterval of $(-\rho,\rho)$. To prove this, we use the classical Beurling-Malliavin completeness theorem and a recent result on density of sign changes of real measures with spectral gap at the origin by M. Mitkovski and A. Poltoratski.
The talk is based on joint work with Ilya Zlotnikov.