论文标题
四角cantor套件的上限和下限
Upper and lower bounds on the rate of decay of the Favard curve length for the four-corner Cantor set
论文作者
论文摘要
平面子集的最爱长度定义为其正交投影的平均值。该数量与概率的Buffon针头问题有关。也就是说，集合的最爱长度与随机掉落到集合的针或线的概率成正比。如果我们掉落固定的曲线，而不是将线路放在集合上，则相关的Buffon曲线概率与所谓的最爱曲线长度成正比。正如我们在同伴论文中显示的那样，一个贝西科维奇的广义投影定理仍然存在于线路被曲线替换的环境中。因此，任何纯粹不可分割的集合的最爱曲线长度为零。由于四角康托尔集是一种紧凑的，纯粹的不可分割的$ 1 $ - 设置，具有有限的，非零的Hausdorff度量，因此其最爱曲线长度等于零。在本文中，我们估算了四角康托尔套件的偏爱曲线长度的衰减速率的上限和下限。我们的技术建立在以前用于经典的最爱长度的想法上。
The Favard length of a subset of the plane is defined as the average of its orthogonal projections. This quantity is related to the probabilistic Buffon needle problem; that is, the Favard length of a set is proportional to the probability that a needle or a line that is dropped at random onto the set will intersect the set. If instead of dropping lines onto a set, we drop fixed curves, then the associated Buffon curve probability is proportional to the so-called Favard curve length. As we show in our companion paper, a Besicovitch generalized projection theorem still holds in the setting where lines are replaced by curves. Consequently, the Favard curve length of any purely unrectifiable set is zero. Since the four-corner Cantor set is a compact, purely unrectifiable $1$-set with bounded, non-zero Hausdorff measure, then its Favard curve length equals zero. In this article, we estimate upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set. Our techniques build on the ideas that have been previously used for the classical Favard length.
