Improved bounds for restricted projection families via weighted Fourier restriction

Abstract

It is shown that if $A\subseteq {\mathbb{R}}^3$ is a Borel set of Hausdorff dimension $\dim A\in (3/2,5/2)$, then for a.e. $\theta \in [0,2\pi )$ the projection ${\pi }_{\theta }(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta ,\sin \theta ,1)$ satisfies $\dim {\pi }_{\theta }(A)\ge \max \left\{\frac{4\dim A}{9}+\frac{5}{6},\frac{2\dim A+1}{3}\right\}$. This improves the bound of Oberlin and Oberlin, and of Orponen and Venieri, for $\dim A\in (3/2,5/2)$. More generally, a weaker lower bound is given for families of planes in ${\mathbb{R}}^3$ parametrised by curves in $S^2$ with nonvanishing geodesic curvature.

50 pages, 1 figure. Accepted version

References in corpus (1)

• Fractal Strichartz estimate for the wave equation

Cited by in corpus (2)

• Restricted families of projections onto planes: The general case of nonvanishing geodesic curvature
• Dimension Distortion by Right Coset Projections in the Heisenberg Group