Rearrangement of vector series. I (Q2709858)

Let \(\mathbb{R}^d_*\) be the one point compactification of the Euclidean space \((d\geq 2)\). Given a permutation \(f\) of the positive integers, let \(C_f (\mathbb{R}^d_*)\) denote the set of all \(C\subset\mathbb{R}^d_*\) for which there is a series \(\sum a_n\) in \(\mathbb{R}^d\) with zero sum such that \(C\) is the cluster set in \(\mathbb{R}^d_*\) of the sequence of partial sums of \(\sum a_{f(n)}\). Every \(C\) in \(C_f(\mathbb{R}^d_*)\) is a non-empty, connected and closed set in \(\mathbb{R}^d_*\). The authors give a combinatorial characterization of the permutations \(f\) for which all non-empty, closed and connected subsets of \(\mathbb{R}^d_*\) belong to \(C_f (\mathbb{R}^d_*)\). For every permutation \(f\) the set \(C\) in \(C_f(\mathbb{R}^d_*)\) which contains ``*'' is determined.