Existence of further uniquely achievable Cantorvals

Determine whether there exist uniquely achievable Cantorvals other than those arising from the sufficient conditions of Theorem \ref{Cantorvale unikalne}.

Background

The paper proves unique achievability for a broad family of Cantorvals satisfying the hypotheses of Theorem \ref{Cantorvale unikalne}, including the Guthrie–Nymann Cantorval and additional examples. It also shows that the Guthrie–Nymann Cantorval is the only uniquely achievable member of a particular family generated by Theorem \ref{mami}. Beyond these constructions, the authors state that their observations are preliminary and leave open whether any other uniquely achievable Cantorvals exist.

References

Are there uniquely achievable Cantorvals other than those arising from Theorem \ref{Cantorvale unikalne}?

— 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, final paragraph; first displayed Problem

Is it true that a Cantorval $E(x_n)$ is uniquely achievable if and only if the sequence $(x_n)$ does not contain neither a duplication of any term, nor a term repeating three times, nor a triple of a term that repeats two times?

— 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, final paragraph; second displayed Problem