Improved bounds for restricted projection families via weighted Fourier restriction

TL;DR

This paper improves bounds on the Hausdorff dimension of projections of sets in three-dimensional space onto planes, using weighted Fourier restriction techniques, extending previous results for a range of set dimensions.

Contribution

It introduces new bounds for the dimension of projections of sets in A3, improving upon prior results, by employing weighted Fourier restriction methods for families of planes parametrized by curves.

Findings

Enhanced lower bounds for projection dimensions in A3 for   (3/2, 5/2)

Extension of bounds to families of planes parametrized by curves with nonvanishing geodesic curvature

Improved theoretical understanding of Fourier restriction in geometric measure theory

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 $\dim \pi_{\theta}(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. 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.