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An estimate from below for the Buffon needle probability of the four-corner Cantor set | | Michael Bateman
; Alexander Volberg
; | | Date: |
18 Jul 2008 | | Abstract: | Let $Cant_n$ be the $n$-th generation in the construction of the middle-half
Cantor set. The Cartesian square $K_n = Cant_n imes Cant_n$ 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 $exp(- clog_* n)$, 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 Nazarov-Peres-Volberg paper
(arxiv:math 0801.2942) the power estimate from above was obtained. The exponent
in this paper 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
$frac{c}{n}$. Here we apply the idea from papers of Nets Katz (MRL (1996),
527-536) and Bateman-Katz (arxiv:math/0609187v1 2006) to show that the estimate
from below can be in fact improved to $c frac{log n}{n}$. This is in drastic
difference from the case of {em random} Cantor sets studied by Peres and
Solomyak in Pacific J. Math. 204 (2002), 473-496. | | Source: | arXiv, 0807.2953 | | Services: | Forum | Review | PDF | Favorites | |
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