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

TL;DR

This paper investigates the topological, metric, and fractal characteristics of a family of self-similar sets, revealing their structure as Cantorvals with specific measure and dimension properties.

Contribution

It introduces a new class of self-similar sets, proving they are Cantorvals and calculating their Lebesgue measure and Hausdorff dimension.

Findings

$K_l$ is a Cantorval with non-empty interior

Lebesgue measure of $K_l$ is computed

Hausdorff dimension of the boundary is determined

Abstract

Depending on a natural parameter $l$, we study the topological, metric, and fractal properties of the homogeneous self-similar set $$K_{l}=\left\{\sum_{i=1}^{\infty} \frac{\varepsilon_i}{(2l+2)^i} : (\varepsilon_i) \in \{0, 2, 4, \dots, 2l, 2l+1, 2l+3, \dots, 4l+1 \}^{\mathbb{N}} \right\}.$$ 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.