Topological Methods in Nonlinear Analysis

Topological and measure properties of some self-similar sets

Taras Banakh, Artur Bartoszewicz, Małgorzata Filipczak, and Emilia Szymonik

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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

Permanent link to this document
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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