Multigeometric sequences and Cantorvals

Artur Bartoszewicz; Małgorzata Filipczak; Emilia Szymonik

Open Mathematics (2014)

  • Volume: 12, Issue: 7, page 1000-1007
  • ISSN: 2391-5455

Abstract

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For a sequence x ∈ l 10, one can consider the achievement set E(x) of all subsums of series Σn=1∞ x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞ x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie J.A., Nymann J.E., The topological structure of the set of subsums of an infinite series, Colloq. Math., 1988, 55(2), 323–327] we describe families of sequences which contain, according to our knowledge, all known examples of x with E(x) being Cantorvals.

How to cite

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Artur Bartoszewicz, Małgorzata Filipczak, and Emilia Szymonik. "Multigeometric sequences and Cantorvals." Open Mathematics 12.7 (2014): 1000-1007. <http://eudml.org/doc/269306>.

@article{ArturBartoszewicz2014,
abstract = {For a sequence x ∈ l 10, one can consider the achievement set E(x) of all subsums of series Σn=1∞ x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞ x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie J.A., Nymann J.E., The topological structure of the set of subsums of an infinite series, Colloq. Math., 1988, 55(2), 323–327] we describe families of sequences which contain, according to our knowledge, all known examples of x with E(x) being Cantorvals.},
author = {Artur Bartoszewicz, Małgorzata Filipczak, Emilia Szymonik},
journal = {Open Mathematics},
keywords = {Multigeometric sequence; Achievement set of sequence; M-Cantorval; multigeometric sequence; achievement set of sequence; cantorval},
language = {eng},
number = {7},
pages = {1000-1007},
title = {Multigeometric sequences and Cantorvals},
url = {http://eudml.org/doc/269306},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Artur Bartoszewicz
AU - Małgorzata Filipczak
AU - Emilia Szymonik
TI - Multigeometric sequences and Cantorvals
JO - Open Mathematics
PY - 2014
VL - 12
IS - 7
SP - 1000
EP - 1007
AB - For a sequence x ∈ l 10, one can consider the achievement set E(x) of all subsums of series Σn=1∞ x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞ x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie J.A., Nymann J.E., The topological structure of the set of subsums of an infinite series, Colloq. Math., 1988, 55(2), 323–327] we describe families of sequences which contain, according to our knowledge, all known examples of x with E(x) being Cantorvals.
LA - eng
KW - Multigeometric sequence; Achievement set of sequence; M-Cantorval; multigeometric sequence; achievement set of sequence; cantorval
UR - http://eudml.org/doc/269306
ER -

References

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  10. [10] Nitecki Z., Subsum sets: intervals, Cantor sets, and Cantorvals, preprint available at http://arxiv.org/abs/1106.3779 
  11. [11] Nymann J.E., Sáenz R.A., The topological structure of the set of P-sums of a sequence, Publ. Math. Debrecen, 1997, 50(3–4), 305–316 Zbl0880.11013
  12. [12] Nymann J.E., Sáenz R.A., On a paper of Guthrie and Nymann on subsums of infinite series, Colloq. Math., 2000, 83(1), 1–4 Zbl0989.11007
  13. [13] Passell N., Series as functions on the Cantor set, Abstracts of Papers Presented to the American Mathematical Society, 1982, 3(1), 65 
  14. [14] Vainshtein A.D., Shapiro B.Z., Structure of a set of a-representable numbers, Izv. Vyssh. Uchebn. Zaved. Mat., 1980, 5, 8–11 (in Russian) 

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