中文摘要：

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-Ui≥1Ui是可数集，则E—E0是至多可数集，其中E0=（x∈E｜Dcs（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．

英文摘要：

For a self-similar set E satisfying the open set condition,we prove an interesting result that if there exists an almost everywhere best covering {Ui）∞i=1 of E such that the set E-Ui≥1Ui is countable,then the set E--E0 is at most countable,where E0=（x∈E｜Dcs（E,x）=1）. As an applieation,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].