Algebraic sums of achievable sets involving Cantorvals

TL;DR

This paper investigates the topological properties of algebraic sums of achievement sets, showing how sums of Cantor and Cantorval sets can transition between different topological types, introducing new characterizations and families.

Contribution

It introduces new characterizations of sequences whose achievement sets are Cantorvals and constructs a new family of such sets beyond multigeometric series.

Findings

Sum of a Cantorval with itself remains a Cantorval

Sum of multiple copies can transition from Cantor set to Cantorval to interval

New family of achievement sets not generated by multigeometric series

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.