BUFFON'S NEEDLE ESTIMATES FOR RATIONAL PRODUCT CANTOR SETS.

Let S∞ = A∞ x B∞ be a self-similar product Cantor set in the complex plane, defined via S∞ = Uj=1 L Tj (S∞), where Tj: ℂ → ℂ have the form Tj (z) = 1/L z+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 -neighborhood of S∞, or equivalently (up to constants), its N -th Cantor iteration. We are interested in the asymptotic behavior 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 C N-p/log log N 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, Łaba-Zhai, and Bond-Volberg. [ABSTRACT FROM AUTHOR]