## Topological Methods in Nonlinear Analysis

### Topological and measure properties of some self-similar sets

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 1013-1028.

Dates
First available in Project Euclid: 21 March 2016

https://projecteuclid.org/euclid.tmna/1458588672

Digital Object Identifier
doi:10.12775/TMNA.2015.075

Mathematical Reviews number (MathSciNet)
MR3494981

Zentralblatt MATH identifier
1362.28009

#### Citation

Banakh, Taras; Bartoszewicz, Artur; Filipczak, Małgorzata; Szymonik, Emilia. Topological and measure properties of some self-similar sets. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 1013--1028. doi:10.12775/TMNA.2015.075. https://projecteuclid.org/euclid.tmna/1458588672

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