Rearrangement of vector series. I

Abstract

Let ℝ^d_* = ℝ^d ∪ {[midast ]} be the one-point compactification of Euclidean space ℝ^d and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]_f(ℝ^d_*) denote the set of all sets C ⊆ ℝ^d_* for which there is a series [sum ]a_n in ℝ^d with zero sum such that C is the cluster set in ℝ^d_* of the sequence of partial sums of [sum ]a_f(n). Every C ∈ [Cscr ]_f(ℝ^d_*) is non-empty, connected and closed in ℝ^d_*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝ^d_* belong to [Cscr ]_f(ℝ^d_*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]_f(ℝ^d_*) which contain [midast ].