Improved bounds for restricted projection families via weighted Fourier restriction

It is shown that if $A \subseteq \mathbb{R}^3$ a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $\theta [0,2\pi)$ the projection $\pi_{\theta}(A)$ $A$ onto 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin 1)$ satisfies \pi_{\theta}(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves bound Oberlin and Oberlin, Orponen Venieri, (3/2,5/2)$. More generally, weaker lower given families planes in $\mathbb{R}^3$ parametrised by curves $S^2$ with nonvanishing geodesic curvature.