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

TL;DR

This paper establishes new bounds on the Hausdorff dimension of projections of sets in three-dimensional space onto planes, showing that for sets with dimension up to 1.5, the dimension is preserved for almost every projection in certain directions.

Contribution

The paper extends known results on projection dimensions from sets with dimension at most 1 to sets with dimension up to 1.5, providing improved bounds and partial results for higher dimensions.

Findings

Dimension preservation for projections of sets with dim ≤ 1.5

Improved bounds for sets with dimension > 1.5

Partial results for higher-dimensional sets

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.