$L^p$-integrability of functions with Fourier supports on fractal sets on the moment curve

For $0<\alpha \le 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in ${\mathbb{R}}^d$ such that $N(E,\epsilon )\lesssim {\epsilon }^{-\alpha }$ for $0<\epsilon <1$ where $N(E,\epsilon )$ is the smallest number of $\epsilon$-balls needed to cover $E$. We proved that if $f\in L^p({\mathbb{R}}^d)$ with \begin{align*} 1 \leq p\leq p_α:= \begin{cases} \frac{d^2+d+2α}{2α} & d \geq 3, \frac{4}α &d =2, \end{cases} \end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem.