Multigeometric sequences and Cantorvals

Abstract

For a sequence $x\in l_1\setminus c_{00}$, one can consider the achievement set $E(x)$ of all subsums of series ${\sum }_{n=1}^{\infty }x(n)$. 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 ${\sum }_{n=1}^{\infty }c(n)$ where $c(2n-1)=\frac{3}{4^n}$ and $c(2n)=\frac{2}{4^n}$ (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.

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