Topological and measure properties of some self-similar sets

Taras Banakh, Artur Bartoszewicz, Emilia Szymonik, Małgorzata Filipczak



Given a finite subset $\Sigma \subset \mathbbR$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the
self-similar set $K(\Sigma ;q)=\bigg\\sum\limits_n=0^\infty
a_nq^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, Mor\'an, Jones and others. The aim of the
paper is to unify and deepen existing piecemeal results.


Self-similar set; multigeometric sequence; Cantorval

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