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Multigeometric sequences and Cantorvals

Artur Bartoszewicz;  Małgorzata Filipczak;  Emilia Szymonik — 2014

Open Mathematics

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...

Algebraic and topological properties of some sets in ℓ₁

Taras Banakh;  Artur Bartoszewicz;  Szymon Głąb;  Emilia Szymonik — 2012

Colloquium Mathematicae

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ∑ n = 1 ∞ x ( n ) . Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of ∑ n = 1 ∞ b ( n ) where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable. We...

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