Decoupling and near-optimal restriction estimates for Cantor sets

Abstract

For any $\alpha \in (0,d)$, we construct Cantor sets in ${\mathbb{R}}^d$ of Hausdorff dimension $\alpha$ such that the associated natural measure $\mu$ obeys the restriction estimate $\Vert \widehat{fd\mu }{\Vert }_p\le C_p\Vert f{\Vert }_{L^2(\mu )}$ for all $p>2d/\alpha$. This range is optimal except for the endpoint. This extends the earlier work of Chen-Seeger and Shmerkin-Suomala, where a similar result was obtained by different methods for $\alpha =d/k$ with $k\in \mathbb{N}$. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of $\Lambda (p)$ sets.

21 pages