Stability of shifts, interpolation, and crystalline measures

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