  In \cite{BRS} we have characterized the existance of a non zero function
vanishing on an open set in terms of the decay of it's Fourier transform on the
$d$-dimensional Euclidean space, the $d$-dimensional torus and on connected,
simply connected two step nilpotent Lie groups. In this paper we improved these
results on $\mathbb R^d$ and prove analogus results on connected, simply
connected nilpotent Lie groups.
