On the Visibility of Homogeneous Cantor Sets

The problems associated with the visible set have been explored by various scholars. In this paper, we investigate the Hausdorff dimension and the topological properties of the visible set in relation to the products of homogeneous Cantor sets. To address these issues and establish our results, we employ beta expansion theory, numerical calculations and several technical results from fractal geometry. Our research reveals that the case of the homogeneous Cantor set differs from those of the middle Cantor sets. Furthermore, we identify a critical number that is linked to both the Hausdorff dimension and the topological properties of the visible set.