Topological and measure properties of some self-similar sets

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 $K(\Sigma;q)=\big\{\sum_{n=0}^\infty a_nq^n:(a_n)_{n\in\omega}\in\Sigma^\omega\big\}$, which is the unique compact solution of the equation $K=\Sigma+qK$. The obtained results are applied to studying partial sumsets $E(x)=\big\{\sum_{n=0}^\infty x_n\varepsilon_n:(\varepsilon_n)_{n\in\omega}\in\{0,1\}^\omega\big\}$ of some (multigeometric) sequences $x=(x_n)_{n\in\omega}$.Comment: 10 page