  It is proved that there does not exist any non zero function in $L^p(\R^n)$
with $1\leq p\leq 2n/\alpha$ if its Fourier transform is supported by a set of
finite packing $\alpha$-measure where $0<\alpha<n$. It is shown that the
assertion fails for $p>2n/\alpha$. The result is applied to prove $L^p$
Wiener-Tauberian theorems for $\R^n$ and M(2).

Functional Analysis