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

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}$beany$2$-plane,whichisnotasubspace\mathrm{.}Weprovethatif$K \subset $\mathbb{R}$^{3}$isaBorelsetwith$\dim_{$\mathrm{H}$} K \leq \tfrac{3}{2}$,then$\dim_{$\mathrm{H}$} \pi_{e}(K) = \dim_{$\mathrm{H}$} K$for$$\mathcal{H}$^{1}$almostevery$e \in S^{2} \cap W$,where$$\mathcal{H}$^{1}$denotesthe$1$-dimensionalHausdorffmeasureand$\dim_{$\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 $3\mathrm{/}2$, improving earlier bounds by D. Oberlin and R. Oberlin.