Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.

中文翻译：

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分形正方形的拓扑 HaUSDorff 维数的下界

给定一个整数[公式：见正文]和一个数字集[公式：见正文]，存在一个满足集合方程[公式：见正文]的自相似集合[公式：见正文]。这组[公式：见正文]称为分形正方形。通过研究[公式：见正文]中包含的线段，我们对分形正方形的拓扑豪斯多夫维数给出了较低的估计。此外，我们计算了非平凡连通分量为平行线段的分形正方形的拓扑豪斯多夫维数，并引入拉丁分形正方形来研究分形正方形的拓扑豪斯多夫维数与其豪斯多夫维数重合的问题。