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2014 | 12 | 7 | 1000-1007
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Multigeometric sequences and Cantorvals

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EN
Abstrakty
EN
For a sequence x ∈ l 1\c 00, 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.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
7
Strony
1000-1007
Opis fizyczny
Daty
wydano
2014-07-01
online
2014-04-03
Twórcy
Bibliografia
  • [1] Banakh T., Bartoszewicz A., Głąb Sz., Szymonik E., Algebraic and topological properties of some sets in ℓ 1, Colloq. Math., 2012, 129(1), 75–85 http://dx.doi.org/10.4064/cm129-1-5
  • [2] Cabrelli C.A., Hare K.E., Molter U.M., Sums of Cantor sets, Ergodic Theory Dynam. Systems, 1997, 17(6), 1299–1313 http://dx.doi.org/10.1017/S0143385797097678
  • [3] Ferens C., On the range of purely atomic probability measures, Studia Math., 1984, 77(3), 261–263
  • [4] 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
  • [5] Hornich H., Über beliebige Teilsummen absolut konvergenter Reihen, Monasth. Math. Phys., 1941, 49, 316–320
  • [6] Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521 http://dx.doi.org/10.4169/amer.math.monthly.118.06.508
  • [7] Kakeya S., On the partial sums of an infinite series, The Science Reports of the Tôhoku University, 1914, 3, 159–164
  • [8] Koshi S., Lai H., The ranges of set functions, Hokkaido Math. J., 1981, 10(Special Issue), 348–360
  • [9] Mendes P., Oliveira F., On the topological structure of the arithmetic sum of two Cantor sets, Nonlinearity, 1994, 7(2), 329–343 http://dx.doi.org/10.1088/0951-7715/7/2/002
  • [10] Nitecki Z., Subsum sets: intervals, Cantor sets, and Cantorvals, preprint available at http://arxiv.org/abs/1106.3779
  • [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
  • [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
  • [13] Passell N., Series as functions on the Cantor set, Abstracts of Papers Presented to the American Mathematical Society, 1982, 3(1), 65
  • [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)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0396-4
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