A Marstrand-type restricted projection theorem in ${\mathbb{R}}^3$

Abstract

Marstrand's projection theorem from $1954$ states that if $K\subset {\mathbb{R}}^3$ is an analytic set, then, for ${\mathcal{H}}^2$ almost every $e\in S^2$, the orthogonal projection ${\pi }_e(K)$ of $K$ to the line spanned by $e$ has Hausdorff dimension $\min \{{\dim }_{\mathrm{H}}K,1\}$. This paper contains the following sharper version of Marstrand's theorem. Let $V\subset {\mathbb{R}}^3$ be any $2$-plane, which is not a subspace. Then, for ${\mathcal{H}}^1$ almost every $e\in S^2\cap V$, the projection ${\pi }_e(K)$ has Hausdorff dimension $\min \{{\dim }_{\mathrm{H}}K,1\}$. For $0\le t<{\dim }_{\mathrm{H}}K$, we also prove an upper bound for the Hausdorff dimension of those vectors $e\in S^2\cap V$ with ${\dim }_{\mathrm{H}}{\rho }_e(K)\le t<{\dim }_{\mathrm{H}}K$.

35 pages, 3 figures. v2: incorporated reviewer comments