Buffon’s needle estimates for rational product Cantor sets

Let S∞ = A∞ × B∞ be a self-similar product Cantor set in the complex plane, defined via S∞ = ⋃L j=1 Tj(S∞), where Tj : C→ C have the form Tj(z) = 1 Lz+zj and {z1, . . . , zL} = A+iB for some A,B ⊂ R with |A|, |B| > 1 and |A||B| = L. Let SN be the L−N -neighbourhood of S∞, or equivalently (up to constants), its N -th Cantor iteration. We are interested in the asymptotic behaviour as N → ∞ of the Favard length of SN , defined as the average (with respect to direction) length of its 1-dimensional projections. If the sets A and B are rational and have cardinalities at most 6, then the Favard length of SN is bounded from above by CN−p/ log logN for some p > 0. The same result holds with no restrictions on the size of A and B under certain implicit conditions concerning the generating functions of these sets. This generalizes the earlier results of Nazarov-Perez-Volberg, Laba-Zhai, and Bond-Volberg.