Topological and measure properties of some self-similar sets

Taras Banakh, Artur Bartoszewicz, Emilia Szymonik, Małgorzata Filipczak

DOI: http://dx.doi.org/10.12775/TMNA.2015.075

Abstract


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.

Keywords


Self-similar set; multigeometric sequence; Cantorval

Full Text:

PREVIEW FULL TEXT

References


T. Banakh, A. Bartoszewicz, S. Głab and E. Szymonik, Algebraic and topological properties of some sets in `1, Colloq. Math. 129 (2012), 75–85.

M. Banakiewicz and F. Prus-Wisniowski, M-Cantorvals of Ferens type, in preparation.

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, Positivemeasure 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.

J. E. Nymann and R. A. Sáenz, 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 Sumpmlambda^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 ovdrlinealpha-representable numbers, Izv. Vyssh. Uchebn. Zaved. Matematika. 24 (1980), 8–11.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism