Rearranging series of vectors on a small set

We consider a variation of the Lévy-Steinitz theorem where one rearranges only a small number of terms. In particular, we study the set of obtainable rearrangements of a multidimensional series by a permutation σ:ω→ω such that {n∈ω:σ(n)≠n}∈I for some proper ideal I⊂P(ω).