Abstract
Given a finite subset $\Sigma \subset \mathbb{R}$ and a positive real number $q\le 1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma ;q)=\bigg\{\sum\limits_{n=0}^{\infty }a_{n}q^{n}:(a_{n})_{n\in \omega }\in \Sigma ^{\omega }\bigg\}$, 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, Morán, Jones and others. The aim of the paper is to unify and deepen existing piecemeal results.
Citation
Taras Banakh. Artur Bartoszewicz. Małgorzata Filipczak. Emilia Szymonik. "Topological and measure properties of some self-similar sets." Topol. Methods Nonlinear Anal. 46 (2) 1013 - 1028, 2015. https://doi.org/10.12775/TMNA.2015.075
Information