Topological and measure properties of some self-similar sets

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

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