AN ESTIMATE FROM BELOW FOR THE BUFFON NEEDLE PROBABILITY OF THE FOUR-CORNER CANTOR SET

Michael Bateman, Alexander Volberg

Research output: Contribution to journalArticlepeer-review

Abstract

Let C-n be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square K-n = C-n x 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(-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 [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 logn/n. This is in drastic contrast to the case of random Cantor sets studied in [13].

Original languageEnglish
Pages (from-to)959-967
Number of pages9
JournalMathematical research letters
Volume17
Issue number5
Publication statusPublished - Sep 2010

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