Universal frame set for rational functions

For $(\lambda, \mu) \in \mathbb{R}^2$ define a time-frequency shift operator $\pi_{\lambda, \mu} $ on $L^2(\mathbb{R})$ by the rule
$$\pi_{\lambda, \mu} g (t):= e^{2\pi i \lambda t} g(t - \mu), \quad g\in L^2(\mathbb{R}).$$
Now for a fixed $g \in L^2(\mathbb{R})$ and countable $L \subset \mathbb{R}^2$ we define a Gabor system $\mathcal{G}(g, L)$ as follows:
$$\mathcal{G}(g, L) := \{\pi_{\lambda, \mu} g \mid (\lambda, \mu) \in L\}.$$
The system $\mathcal{G}(g,L)$ is a Gabor frame if for some constants $A, B >0$ one has
\begin{equation} A\|f\|^2_2\leq \sum_{(\lambda, \mu) \in L}|(f, \pi_{\lambda, \mu}g)|^2\leq B\|f\|^2_2, \text{ for any } f\in L^2(\mathbb{R}). \end{equation}

Definition. For any $M \in \mathbb{N}$ let $\mathcal{K}_1(M)$ be a class of rational functions of degree $M$, i.e. it has the form
$$g(t) = \sum_{k=1}^{N} {{a_k} \over {(t - i w_k)^{j_k}}}, \text{ where } a_k \in \mathbb{C}, w_k \in \mathbb{C} \setminus i\mathbb{R} \text{ and } \sum_{k=1}^N j_k = M,$$
such that
\begin{equation}\label{eq:defK2} \sum_{k=1}^N a_k e^{2\pi w_k t} {{(2\pi i)^{j_k-1}} \over {(j_k-1)!}} t^{j_k-1} \ne 0 \text{ for any } t <0. \end{equation}

For example, if all the poles of $g$ lie in the upper half-plane, then (\ref{eq:defK2}) is equivalent to the simple condition
\begin{equation} \widehat{g}(t) \ne 0 \text{ for any } t >0. \end{equation}

Definition. For a set $L$ its upper density $D(\Lambda)$ is defined by the formula
$$D(\Lambda) = \lim_{a \to \infty} \sup_{R \in \mathbb{R}} {{\# \{x \in \Lambda \mid x \in [R, R+a]\}} \over {a}}.$$

In the talk we discuss the following universal result:
Theorem 1. For any $\varepsilon >0$ and any $M \in \mathbb{N}$ there exist a set $\Lambda = \Lambda(\varepsilon, M) \subset \mathbb{R}$ of density $D(\Lambda) \leq 1+\varepsilon$ such that the system
$$\mathcal{G}(g, \Lambda\times \mathbb{Z}) := \{e^{2\pi i \lambda t} g(t - n) \mid (\lambda, n) \in \Lambda \times \mathbb{Z}\}$$
is a frame in $L^2(\mathbb{R})$ for any rational function $g \in \mathcal{K}(M)$.