贝西科维奇集的豪斯道夫测度为无穷大  

The Besicovitch Sets Have Infinite Hausdorff Measures

作　　者：马际华[1] 

机构地区：[1]武汉大学数学科学学院/非线性科学中心,湖北武汉430072

出　　处：《武汉大学学报（自然科学版）》2001年第1期6-8,共3页Journal of Wuhan University(Natural Science Edition)

基　　金：国家自然科学基金!资助项目 (2 0 2 0 0 0 1 33)

摘　　要：给定一个概率向量P =(p0 ,p1,… ,pm -1) (m≥ 2 ) ,贝西科维奇集B由单位区间中那些在m 进制展开 ,式中j(0≤j≤m - 1)出现的频率为pj((0≤j≤m - 1) )的点组成 ,已经知道它在任何量纲下的豪斯道夫测度非零即无穷 本文运用测度的微扰法证明了西科维奇集的豪斯道夫测度为无穷大 .更进一步 ,证明了西科维奇集在量纲h(t) =tsexp{ -c |logt|log|logt| }之下的豪斯道夫测度为无穷大 .For a fixed probability vector P=(p_0,p_1,…,p_ m-1)(m≥2),The Besicovitch set B is the set of points in the unit interval which contain j(0≤j≤m-1) in their madic expansions in the propotion p_j((0≤j≤m-1)); it is known that B has Hausodrff dimension-∑p_jlogp_jlogm, and its Hausdorff measures(under any gauge) are either zero or infinity. A kind of peturbation measures are employed to prove that the Besicovitch set has infinite Hausdorff measure in the dimension.

关 键 词：贝西科维奇集 豪斯道夫测度 量纲 概率向量 微扰测度法 概率测度 无穷大 

分 类 号：O174.12[理学—数学]