An estimate from below for the Buffon needle probability of the four-corner Cantor set

Abstract

Let Cn be the n-th generation in the construction of the middlehalf Cantor set. The Cartesian square Kn = Cn × Cn 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 Kn is essentially the average length of the projections of Kn, also known as the Favard length of Kn. A classical theorem of Besicovitch implies that the Favard length of Kn tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper boundwas exp(−c log * n), due to Peres and Solomyak. (log * n is the number of times one needs to take log to obtain a number less than 1 starting from n). In [11] the power estimate from above was obtained. The exponent in [11] was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate c n . Here we apply the idea from [4], [1] to show that the estimate from below can be in fact improved to c log n n . This is in drastic difference from the case of random Cantor sets studied in [13].

Circular Favard Length of the Four-Corner Cantor Set

“…These almost-disjoint projected intervals induce a 4-adic structure on the interval. Since 0 is a direction with a large projection, the idea is that as θ gets exponentially farther from 1/n ≈ 0, the average projections of K n should shrink-but [3] shows that this happens no more quickly than the same exponential rate.…”

“…We will describe the distortion and show that for r large enough, K n is sufficiently coarse for the argument of [3] to yield the same lower bound.…”

“…Let us review briefly the argument of [3] which proves that Fav(K n ) ≥ C log n n . Below, we will fix an n, and none of the constants will depend on n. We rotate the axes, defining θ = 0 to be the direction arctan(1/2), because K n projects onto this direction nicely: the projected squares together fill out a single connected interval, and the projected squares intersect only on their endpoints.…”

“…This can be slightly improved, but it is still a long way till τ = 1. It was shown in [3], using the idea of [1,2], that τ = 1 is impossible.…”

The Favard length of product Cantor sets

“…This property also turns out to be crucial for the proof of the lower bound in [1]. We expect this to be the "worst" case in the sense that if all projections of a 1-dimensional Cantor set E ∞ have 1-dimensional measure 0, then Mattila's lower bound should be sharp, i.e.…”