论文标题
Dirichlet不仅在许多理性的IF分形中都是糟糕和奇异的
Dirichlet is not just bad and singular in many rational IFS fractals
论文作者
论文摘要
对于$ m \ ge 2 $，请考虑$ k $ $ m $ $倍的笛卡尔产品，其中有两个具有理性系数的IFS的限制集。如果IFS的收缩率是整数的倒数，而$ k $并不归入Singleton，那么我们以$ k $的构建向量，这些向量位于Beresnevich等人所定义的``民间传说集合''中，这意味着它们是可以透明的，但不是很奇怪，但不是很奇怪，但实际上是近似的（实际上我们的例子）。我们进一步介绍了hausdorff的下限和这些民间传说集的包装维度的主题，但是我们没有明确计算界限。我们的一类分形扩展了经典数字缺失分形的（笛卡尔产物），最近获得了类似的结果。
For $m\ge 2$, consider $K$ the $m$-fold Cartesian product of the limit set of an IFS of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and $K$ does not degenerate to singleton, we construct vectors in $K$ that lie within the ``folklore set'' as defined by Beresnevich et al., meaning they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within $K$, however we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.
