On the packing dimension of Furstenberg sets

We prove that if $\alpha\in (0,1/2]$, then the packing dimension of a set $E\subset\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\ge \alpha$ is at least $1/2+\alpha+c(\alpha)$ for some $c(\alpha)>0$. In particular, this holds for $\alpha$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $\ge\alpha$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $\alpha$-Furstenberg sets over the "trivial" estimate for all values of $\alpha\in (0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $\alpha$-Furstenberg set resembles a set of dimension close to $1/2+\alpha$, if they exist, are rather sparse.
Comment: 12 pages. v2: incorporates referee's comments, to appear in Journal d'Analyse Math\'ematique