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In \cite{BRS} we have characterized the existance of a non zero functionvanishing 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 theseresults on $\mathbb R^d$ and prove analogus results on connected, simplyconnected nilpotent Lie groups.

Improved Ingham-type result on $\mathbb R^d$ and on connected, simply connected nilpotent Lie Groups