  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).
