Density of complete and minimal systems of time frequency shifts of Gaussians

It was shown by Ascenzi, Lyubarskii, Seip that the upper density of any complete and minimal system of Gaussians is between $\frac{2}{\pi}$ and 1, if we know that there exists an angular density. It was an open question whether it is true in general. We will show that there exists a complete and minimal system with upper density $\dfrac{1}{\pi}$, and that the uppper density of any such system is larger than $\dfrac{1}{3\pi}$.
The talk is based on work in collaboration with A. Borichev and A. Kuznetsov.