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.
Categories: math.CA