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arXiv · 2309.01589

Algebraic sums of achievable sets involving Cantorvals

Jacek Marchwicki·Piotr Nowakowski·Franciszek Prus-Wiśniowski
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Abstract

In this paper we look at the topological type of algebraic sum of achievement sets. 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 for any $m,p \in (\mathbb{N}\setminus \{1\}) \cup \{\infty\}$, $p \geq m$, the algebraic sum of $k$ copies of a Cantor set can transit from a Cantor set to a Cantorval for $k=m$ and then to an interval for $k=p$. These two main results are based on a new characterization of sequences whose achievement sets are Cantorvals. We also define a new family of achievable Cantorvals which are not generated by multigeometric series. In the final section we discuss various decompositions of sequences related to the topological typology of achievement sets.

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Jacek Marchwicki, Piotr Nowakowski, Franciszek Prus-Wiśniowski. 2023-09-04. Algebraic sums of achievable sets involving Cantorvals. https://arxiv.org/abs/2309.01589

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