From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that linear combinations of translates of f \in L^2(R) are dense in L^2 if and only if the Fourier transform of f is nonzero almost everywhere. Is there a characterization (in terms of the Fourier transform) of functions in L^p(R) with the property that linear combinations of its translates are dense in L^p?

If the answer is no can it be shown that no reasonable measure of the size of the zero set of the Fourier transform of f will suffice to give such a characterization?