THE POWER LAW FOR THE BUFFON NEEDLE PROBABILITY OF THE FOUR-CORNER CANTOR SET
THE POWER LAW FOR THE BUFFON NEEDLE PROBABILITY OF THE FOUR-CORNER CANTOR SET
复制标题
四角康托集布冯针概率的幂律
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
A. Volberg
中科院分区：
文献类型：
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作者：
F. Nazarov;Y. Peres;A. Volberg
设Cn是构造中半康托集的第n代。Cn的笛卡尔平方Kn由4n个边长为4 - n的平方组成。在单位方格中随机掷出的长针遇到Kn的概率本质上是Kn投影的平均长度，也称为Kn的法瓦德长度。贝西科维奇的一个经典定理表明，Kn的法瓦德长度趋于零。确定它的确切衰变速率仍是一个悬而未决的问题。直到最近，由于Peres和Solomyak，唯一明确的上界是exp（c log * n）。（log * n是需要取log以从n开始得到小于1的数的次数）。我们结合分析和组合的思想得到幂律界。
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 −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 bound was 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). We obtain a power law bound by combining analytic and combinatorial ideas.