Topological Methods in Nonlinear Analysis

Topological and measure properties of some self-similar sets

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 21 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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.

Export citation


  • A. Bartoszewicz, M. Filipczak and E. Szymonik, Multigeometric sequences and Cantorvals, Cent. Eur. J. Math. 12 (2014), 1000–1007.
  • M. Cörnyei, T. Jordan, M. Pollicott, D. Preiss and B. Solomyak, Positive-measure self-similar sets without interior, Ergodic Theory Dynam. Systems. 26 (2006), 755–758.
  • C. Ferens, On the range of purely atomic probability measures, Studia Math. 77 (1984), 261–263.
  • J.A. Guthrie and J.E. Nymann, The topological structure of the set of subsums of an infinite series, Colloq. Math. 55 (1988), 323–327.
  • R. Jones, Achievement sets of sequences, Amer. Math. Monthly 118 (2011), 508–521.
  • S. Kakeya, On the partial sums of an infinite series, Tôhoku Sci. Rep. 3 (1914), 159–164.
  • P. Mendes and F. Oliveira, On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity 7 (1994), 329–343.
  • M. Morán, Fractal series, Mathematika 36 (1989), 334–348.
  • J.E. Nymann and R.A. Sáenz, The topological structure of the sets of $P$-sums of a sequence II, Publ. Math. Debrecen. 56 (2000), 77–85.
  • ––––, On the paper of Guthrie and Nymann on subsums of infinite series, Colloq. Math. 83 (2000), 1–4.
  • A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111–115.
  • B. Solomyak, On the random series $\Sigma \pm \lambda ^{n}$ (an Erdös problem), Ann. Math. 142 (1995), 611–625.
  • H. Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fund. Math. 1 (1920), 93–104.
  • A.D. Weinstein and B.E. Shapiro, On the structure of a set of $\overline{\alpha }$-representable numbers, Izv. Vyssh. Uchebn. Zaved. Matematika. 24 (1980), 8–11.