Near-optimal restriction estimates for Cantor sets on the parabola

TL;DR

This paper constructs Cantor sets on the parabola with Hausdorff dimension lpha, which are Salem sets satisfying near-optimal restriction estimates for Fourier transforms, extending previous work on fractal sets to a curved setting.

Contribution

It introduces a method to build Cantor sets on the parabola that achieve near-optimal restriction estimates, adapting techniques from fractal measures in Euclidean space.

Findings

Constructed Salem sets on the parabola with specified Hausdorff dimension.

Established Fourier restriction estimates for these sets for all p > 6/lpha.

Extended restriction estimate techniques from Euclidean fractals to curved fractal sets.

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 $\|{\widehat{f d\nu}}\|_{L^p(\mathbb{R}^2)} \leq C_p \|{f}\|_{L^2(\nu)}$ 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 Laba 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.