The center of distances of central Cantor sets (Q6078324)

The center of distances \(S(Y)\) of a non-empty subset \(Y\) of a metric space \((X,d)\) is the set of all real numbers \(\alpha \in [0, +\infty)\) such that for any \(x\in X\) there exists \(y\in X\) with \(d(x,y)=\alpha\). This notion was introduced and pre-tested in [\textit{W. Bielas} et al., Eur. J. Math. 4, No. 2, 687--698 (2018; Zbl 1422.40001)]. Then, it was considered in several other papers, including the one by \textit{M. Banakiewicz} et al. [Result. Math. 77, No. 5, Paper No. 196, 20 p. (2022; Zbl 1504.54021)]. The scope of the continuation is best reflected by summary fragments. Namely, a ``central Cantor set is an achievement set of exactly one fast convergent sequence \((a_n)\). We show that the center of distances of the central Cantor set is the union of all centers of distances of \(F_n\), where \(F_n\) is the set of all \(n\)-initial subsums of the series \(\sum a_n\). Moreover, we give a necessary and sufficient condition for central Cantor sets to have not the minimal possible center of distances.''