【摘要】 对于一个满足开集条件的自相似集E,本文得到如下有趣结论:如果E存在几乎处处最好覆盖{Ui}∞i=1,使得E-∪i≥1Ui是可数集,则E-E0是至多可数集,其中E0={x∈E|珡Ds c(E,x)=1}.作为应用,否定回答了周作领等在[周作领,瞿成勤,朱智伟.自相似集的结构———Hausdorff测度与上凸密度[M].北京:科学出版社,2008]中提出的一个公开问题.更多还原

【Abstract】 For a self-similar set Esatisfying the open set condition,we prove an interesting result that if there exists an almost everywhere best covering{Ui}∞i=1 of Esuch that the set E-∪i≥1Uiis countable,then the set E-E0is at most countable,where E0= {x∈E|Dsc(E,x)=1}.As an application,we give a negative answer to the open problem posed by ZHOU Zuoling et.al in[ZHOU Zuoling,QU Chengqin,ZHU Zhiwei.The Structure of Self-similar Sets-Hausdorff Measure and Upper Convex Density.Beijing:Sci & Tec Press,2008].更多还原