Near-Optimal Restriction Estimates for Cantor Sets on the Parabola

Abstract

Abstract For any $0 < \alpha <1$, we construct Cantor sets on the parabola of Hausdorff dimension $\alpha $ such that they are Salem sets and each associated measure $\nu $ satisfies the estimate $\left \| \widehat {f d\nu } \right \|_{L^{p}({\mathbb {R}}^{2})} \leq C_{p} \left \| f \right \|_{L^{2}(\nu )}$ for all $p>6/\alpha $ and for some constant $C_{p}>0$,which may depend on $p$ and $\nu $. The range $p>6/\alpha $ is optimal except for the endpoint. The proof is based on the work of Łaba and Wang on restriction estimates for random Cantor sets and the work of Shmerkin and Suomala on Fourier decay of measures on random Cantor sets. They considered fractal subsets of ${\mathbb {R}}^{d}$, while we consider fractal subsets of the parabola.