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In this paper we look at the topological type of algebraic sum of achievementsets. We show that there is a Cantorval such that the algebraic sum of its $k$copies is still a Cantorval for any $k \in \mathbb{N}$. We also prove that forany $m,p \in (\mathbb{N}\setminus \{1\}) \cup \{\infty\}$, $p \geq m$, thealgebraic sum of $k$ copies of a Cantor set can transit from a Cantor set to aCantorval for $k=m$ and then to an interval for $k=p$. These two main resultsare based on a new characterization of sequences whose achievement sets areCantorvals. We also define a new family of achievable Cantorvals which are notgenerated by multigeometric series. In the final section we discuss variousdecompositions of sequences related to the topological typology of achievementsets.

Algebraic sums of achievable sets involving Cantorvals