Title: 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 Järvenpää,
Järvenpää, 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.