Topological types of two explicitly defined achievement sets

Determine the topological type of the achievement sets generated by the sequences b_n=1/2^n+(-1)^n/3^n and a_{2n-1}=1/2^{2n-1}, a_{2n}=1/(2^{2n}+1).

Background

The paper gives two non-multigeometric sequences whose achievement sets are proved to be uniquely achievable by the preceding Cantorval criterion. However, unique achievability does not determine whether an achievement set is a Cantor set or a Cantorval. The authors explicitly identify the topological classification of both examples as unresolved and describe the problem as challenging.

References

The paradox is that we do not know the topological type of either of these two achievement sets, and this problem appears to be quite challenging (cf. Problems 1.8 and 1.9).

— A characterization of uniqueness of purely atomic finite measures with central Cantor set range  (2608.14395 - Nowakowski et al., 14 Aug 2026) in Section 3, paragraph beginning “There are also two particular non-multigeometric sequences”