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 $\dim_{\mathrm{H}} K \leq \tfrac{3}{2}$, 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 J\"arvenp\"a\"a, J\"arvenp\"a\"a, Ledrappier and Leikas, for Borel sets $K \subset \mathbb{R}^{3}$ with $\dim_{\mathrm{H}} K \leq 1$. We also prove a partial result for sets with dimension exceeding $3/2$, improving earlier bounds by D. Oberlin and R. Oberlin.