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.

中文翻译：

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$\mathbb{R}^{3}$ 中平面投影受限族的改进边界

对于 $e \in S^{2}$，$\mathbb{R}^3$ 中的单位球面，令 $\pi_{e}$ 为 $e^{\perp} \subset \mathbb 的正交投影{R}^{3}$，让 $W \subset \mathbb{R}^{3}$ 是任何 $2$-plane，它不是一个子空间。我们证明如果 $K \subset \mathbb{R}^{3}$ 是一个带有 $\dim_{\mathrm{H}} K \leq \tfrac{3}{2}$ 的 Borel 集，那么 $\dim_ {\mathrm{H}} \pi_{e}(K) = \dim_{\mathrm{H}} K$ for $\mathcal{H}^{1}$ 几乎每个 $e \in S^{2} \cap W$，其中$\mathcal{H}^{1}$ 表示$1$ 维Hausdorff 测度，$\dim_{\mathrm{H}}$ 表示Hausdorff 维。这是早先知道的，由于 J\"arvenp\"a\"a, J\"arvenp\"a\"a, Ledrappier 和 Leikas，对于 Borel 集 $K \subset \mathbb{R}^{3}$与 $\dim_{\mathrm{H}} K \leq 1$。我们还证明了维度超过 $3/2$ 的集合的部分结果，通过 D 改进了早期的界限。