Topological and measure properties of some self-similar sets

Abstract

Given a finite subset $\Sigma \subset \mathbb{R}$ and a positive real number $q\le 1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma ;q)=\bigg\{\sum\limits_{n=0}^{\infty }a_{n}q^{n}:(a_{n})_{n\in \omega }\in \Sigma ^{\omega }\bigg\}$, which is the unique compact solution of the equation $K=\Sigma +qK$. The obtained results are applied to studying partial sumsets $E(x)=\bigg\{\sum\limits_{n=0}^{\infty }x_{n}\varepsilon _{n}:(\varepsilon _{n})_{n\in \omega }\in \{0,1\}^{\omega } \bigg\}$ of multigeometric sequences $x=(x_{n})_{n\in \omega }$. Such sets were investigated by Ferens, Morán, Jones and others. The aim of the paper is to unify and deepen existing piecemeal results.