On the packing dimension of Furstenberg sets

We prove that if alpha is an element of (0, 1/2], then the packing dimension of a set E subset of Double-struck capital R-2 for which there exists a set of lines of dimension 1 intersecting E in Hausdorff dimension >= 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 >= >= 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 is an element of (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.