Discrete Translates in Function Spaces

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