Stability of shifts, interpolation, and crystalline measures

Let $V_{\mathrm{\Gamma }}^p(\mathcal{G}),1\le p\le \mathrm{\infty },$ be the quasi shift-invariant space generated by $\mathrm{\Gamma }$-shifts of a function $\mathcal{G}$, where $\mathrm{\Gamma }\subset \mathbb{R}$ is a separated set. For several large families of generators $\mathcal{G}$, we present necessary and sufficient conditions on $\mathrm{\Gamma }$ that imply that the $\mathrm{\Gamma }$-shifts of $\mathcal{G}$ form an unconditional basis for $V_{\mathrm{\Gamma }}^p(\mathcal{G})$. The connection between this property, interpolation, universal interpolation, and crystalline measures is discussed.