Topological, metric and fractal properties of one family of self-similar sets

Abstract

Depending on a natural parameter $l$, we study the topological, metric, and fractal properties of the homogeneous self-similar set $$K_l=\left\{\sum _{i=1}^{\infty }\frac{{\epsilon }_i}{(2l+2{)}^i}:({\epsilon }_i)\in \{0,2,4,\dots ,2l,2l+1,2l+3,\dots ,4l+1{\}}^{\mathbb{N}}\right\}.$$ In particular, we prove that $K_l$ is a Cantorval, that is, a perfect set on the real line with a non-empty interior and fractal boundary. Additionally, we compute the Lebesgue measure of $K_l$ and the Hausdorff dimension of its boundary.