The power law for the Buffon needle probability of the four-corner Cantor set

Abstract

Let $C_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $K_n$ is essentially the average length of the projections of $K_n$, also known as the Favard length of $K_n$. A classical theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp (-c{\log }_{\ast }n)$, due to Peres and Solomyak. (${\log }_{\ast }n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). We obtain a power law bound by combining analytic and combinatorial ideas.

16 pages, 2 figures

References in corpus (1)

• Transport and mixing by internal waves in stellar interiors: effect of the Coriolis force

Cited by in corpus (3)

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