Minimal centers of distances for semi-fast convergent sequences

Characterize the semi-fast convergent sequences whose achievement sets have minimal centers of distances.

Background

The paper studies unique achievability of symmetric Cantor sets represented by semi-fast convergent sequences with multiplicities. Its proof cannot use the same center-of-distances method employed for central Cantor sets because no criterion for minimality of the center of distances is available in the semi-fast setting. A characterization of precisely which semi-fast convergent sequences have minimal centers of distances would provide the missing structural result and could support extensions of the uniqueness arguments.

References

Finding such a characterization remains an interesting open problem.

\begin{problem} Characterize the semi-fast convergent sequences that have minimal centers of distances. \end{problem}

— A characterization of uniqueness of purely atomic finite measures with central Cantor set range  (2608.14395 - Nowakowski et al., 14 Aug 2026) in Section 2, immediately after the proof of Theorem 5; Problem statement