Improved bounds for restricted families of projections to planes in ${\mathbb{R}}^3$

Abstract

For $e\in S^2$, the unit sphere in ${\mathbb{R}}^3$, let ${\pi }_e$ be the orthogonal projection to $e^{\perp }\subset {\mathbb{R}}^3$, and let $W\subset {\mathbb{R}}^3$ be any $2$-plane, which is not a subspace. We prove that if $K\subset {\mathbb{R}}^3$ is a Borel set with ${\dim }_{\mathrm{H}}K\le \frac{3}{2}$, then ${\dim }_{\mathrm{H}}{\pi }_e(K)={\dim }_{\mathrm{H}}K$ for ${\mathcal{H}}^1$ almost every $e\in S^2\cap W$, where ${\mathcal{H}}^1$ denotes the $1$-dimensional Hausdorff measure and ${\dim }_{\mathrm{H}}$ the Hausdorff dimension. This was known earlier, due to Järvenpää, Järvenpää, Ledrappier and Leikas, for Borel sets $K\subset {\mathbb{R}}^3$ with ${\dim }_{\mathrm{H}}K\le 1$. We also prove a partial result for sets with dimension exceeding $3/2$, improving earlier bounds by D. Oberlin and R. Oberlin.

12 pages. v2: minor corrections

References in corpus (2)

• On restricted families of projections in R^3
• A Marstrand-type restricted projection theorem in ${\mathbb{R}}^3$

Cited by in corpus (1)

• Improved bounds for restricted projection families via weighted Fourier restriction