Given a Gabor orthonormal basis of $L^2(\mathbb{R})$\[\mathcal{G}(g,T,S):=\big\{ g(x-t) e^{2\pi is x}: g\in L^2(\mathbb{R}), \,t\in T,\, s\in S\big\},\]we study periodicity properties of the translation and modulation sets $T$ and $S$. In particular, we show that if the window function $g$ is compactly supported, then $T$ and $S$ must be periodic sets, i.e., of the form\[T = a\mathbb{Z}+ \{t_1,\ldots,t_n\}, \qquad S = b\mathbb{Z} + \{s_1,\ldots,s_m\}.\]To achieve this, we first obtain a result of independent interest: if the system $\mathcal{G}(g,T,S)$ is an orthonormal basis of $L^2(\mathbb{R})$, then both $|g|^2$ and $|\widehat{g}|^2$ tile $\mathbb{R}$ by translations (when translated along the sets $T$ and $S$, respectively), and moreover,\[\sum_{t\in T} |g(x-t)|^2=D(T), \qquad \sum_{s\in S} |\widehat{g}(x-s)|^2=D(S), \qquad \text{a.e. }x\in \mathbb{R},\]where $D(\Lambda)$ denotes the uniform density of a set $\Lambda\subset \mathbb{R}$.Partial results towards the Liu-Wang conjecture are also obtained.

This talked in based on a joint work with Nir Lev (Bar-Ilan University, Israel)

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