Lp-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 $R^d$ such that $N(E,\epsilon )\lesssim {\epsilon }^{-\alpha }$ for $0<\epsilon <1$ where $N(E,\epsilon )$ is the smallest number of ε-balls needed to cover E. We proved that if $f\in L^p\left(R^d\right)$ with$1\le p\le p_{\alpha }:=\{\begin{array}{cc}\frac{d^2+d+2\alpha }{2\alpha }& d\ge 3,\\ \frac{4}{\alpha }& d=2,\end{array}$ and $\hat{f}$ 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 [11], including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem.