Multigeometric sequences and Cantorvals

2013/04/11 by Bartoszewicz, Artur, Filipczak, Małgorzata, Szymonik, Emilia · 4 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.4218

Abstract

For a sequence x ∈ l₁ ∖ c₀₀, one can consider the achievement set E(x) of all subsums of series ∑_n=1^∞ x(n). It is known that E(x) is one of the following types of sets: * finite union of closed intervals, * homeomorphic to the Cantor set, * homeomorphic to the set T of subsums of ∑_n=1^∞ c(n) where c(2n-1)=(3)/(4ⁿ) and c(2n)=(2)/(4ⁿ) (Cantorval). Based on ideas of Jones and Velleman, and Guthrie and Nymann we describe families of sequences which contain, according to our knowledge, all known examples of x's with E(x) being Cantorvals.

Cited by

• On a new condition implying that an achievement set is a Cantorval and its applications
• Achievement sets -- current results and open problems
• The Lebesgue measure of boundaries of multigeometric Cantorvals
• Bounded ranges of cardinal functions

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