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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0396-4,
author = {Artur Bartoszewicz and Ma\l gorzata Filipczak and Emilia Szymonik},
title = {Multigeometric sequences and Cantorvals},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1000-1007},
zbl = {1298.40001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0396-4}
}
Artur Bartoszewicz; Małgorzata Filipczak; Emilia Szymonik. Multigeometric sequences and Cantorvals. Open Mathematics, Tome 12 (2014) pp. 1000-1007. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0396-4/
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