Discrete Translates in Function Spaces

Abstract: We construct a Schwartz function $\varphi$ such that for every exponentially small perturbation of integers $\Lambda$, the set of translates ${\varphi(t-\lambda), \lambda\in\Lambda}$ spans the space $L^p(R)$, for every $p > 1$. This result remains true for more general function spaces $X$, whose norm is "weaker" than $L^1$ (on bounded functions).