Codes and Expansions (CodEx) Seminar

Donggeun Ryou (University of Rochester):
Near-Optimal Restriction Estimates for Cantor Sets on the Parabola

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. This is an analogue of works of Shmerkin-Suomala and Łaba-Wang. They considered fractal subsets of \(\mathbb{R}^{d}\), whereas we consider fractal subsets of the parabola.