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.

arXiv comment50 pages, 1 figure. Accepted version

References in corpus (11)

• On restricted families of projections in R^3
• On the dimension and smoothness of radial projections
• Application of a Fourier restriction theorem to certain families of projections in R^3
• Fractal Strichartz estimate for the wave equation
• On the Hausdorff dimension of pinned distance sets
• Hausdorff dimension and non-degenerate families of projections
• A Marstrand-type restricted projection theorem in ${\mathbb{R}}^3$
• Improved decay of conical averages of the Fourier transform
• An a.e. lower bound for Hausdorff dimension under vertical projections in the Heisenberg group
• Improved bounds for restricted families of projections to planes in ${\mathbb{R}}^3$
• Refined Strichartz inequalities 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