Topological and measure properties of some self-similar sets - Banakh, Taras; Bartoszewicz, Artur; Szymonik, Emilia; Filipczak, Małgorzata (Topological Methods in Nonlinear Analysis, 2015)
Given a finite subset Σ and a positive real number q<1 we study topological and measure-theoretic properties of the self-similar set K(Σ ;q)=sum_n=0^∞ a_nq^n:(a_n)_nω Σ ^ω $, which is the unique compact solution of the equation K=Σ +qK. The obtained results are applied to studying partial sumsets E(x)=sum_n=0^∞ x_nε _n:(ε _n)_nω \0,1ômega % $ of multigeometric sequences x=(x_n)_nω . Such sets were investigated by Ferens, Mor\’an, Jones and others. The aim of the paper is to unify and deepen existing piecemeal results.