Discrete Translates in Function Spaces

We construct a Schwartz function $\phi$ such that for every exponentially small perturbation of integers $\mathrm{\Lambda }$, the set of translates $\{\phi (t-\lambda ),\lambda \in \mathrm{\Lambda }\}$ 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).