Beurling density theorems for sampling and interpolation on the flat cylinder

We consider the Fock space weighted by $e^{-\alpha \mathrm{\mid }z{\mathrm{\mid }}^2}$, of entire and quasi-periodic (modulo a weight dependent on $\nu$) functions on $C$. The quotient space $\mathbb{C}\mathrm{/}\mathbb{Z}$, called The flat cylinder', is represented by the vertical strip $[0,1)\times \mathbb{R}$, which tiles ${C}$ by ${Z}$-translations and is therefore a fundamental domain for $\mathbb{C}/\mathbb{Z}$. Our main result gives a complete characterization of the sets $Z\subset Λ\left( \mathbb{Z}\right) $ that are sets of sampling or interpolation, in terms of concepts of upper and lower Beurling densities, $ D^{+}(Z)$ and $D^{-}(Z)$, adapted to the geometry of $\mathbb{C}/\mathbb{Z}$. The critical Nyquist density’ is the real number $\frac{\alpha }{\pi }$, meaning that the condition $D^{-}(Z)>\frac{\alpha }{\pi }$ characterizes sets of sampling, while the condition $D^{+}(Z)<\frac{\alpha }{\pi }$ characterizes sets of interpolation. The results can be reframed as a complete characterization of Gabor frames and Riesz basic sequences (given by arbitrary discrete sets in $Z\subset \mathrm{\Lambda }\left(\mathbb{Z}\right)$), with time-periodized Gaussian windows (theta-Gaussian), for spaces of functions $f$, measurable in $\mathbb{R}$, square-integrable in $(0,1)$, and quasi-periodic with respect to integer translations.