Topological and measure properties of some self-similar sets

Abstract

Given a finite subset $\Sigma \subset \mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma ;q)=\{{\sum }_{n=0}^{\infty }{a}_n{q}^n:(a_n{)}_{n\in \omega }\in {\Sigma }^{\omega }\}$, which is the unique compact solution of the equation $K=\Sigma +qK$. The obtained results are applied to studying partial sumsets $E(x)=\{{\sum }_{n=0}^{\infty }{x}_n{\epsilon }_n:({\epsilon }_n{)}_{n\in \omega }\in \{0,1{\}}^{\omega }\}$ of some (multigeometric) sequences $x=(x_n{)}_{n\in \omega }$.

arXiv comment10 pages