  We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that
$\widehat{f}$ is supported on a small perturbation of the moment curve in
$\mathbb{R}^k$, then $f$ is identically zero. This improves the more general
result of Agranovsky and Narayanan, and the exponents are sharp in all
dimensions. In the process, we develop a mechanism that should lead to further
progress on related problems.
