The Favard length of product Cantor sets

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...

Recent progress on Favard length estimates for planar Cantor sets

Intersections of Large-radius Circles with the Four-corner Cantor Set: Estimates from below of the Buffon Noodle Probability for Undercooked Noodles

Let Cn be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square Kn of Cn consists of 4 n squares of side-length 4. The chance that a long needle thrown at random in the unit square will meet Kn is essentially the average length of the projections of Kn, also known as the Favard length of Kn. A result due to Bateman and Volberg [1] shows that a lower estimat...

The approximate solutions of Fredholm integral equations on Cantor sets within local fractional operators

In this paper, we apply the local fractional Adomian decomposition and variational iteration methods to obtain the analytic approximate solutions of Fredholm integral equations of the second kind within local fractional derivative operators. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools ...

Products of Special Sets of Real Numbers

We develop a machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: (1) The product of meager/null-additive sets in the Cantor space is meager/nulladditive, respectively. (2) The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measur...