Levy-Steinitz theorem and achievement sets of conditionally convergent series on the real plane

Abstract: Levy-Steinitz theorem characterize sum range of conditionally convergent series, that is a set of all its convergent rearrangements; in finitely dimensional spaces -- it is an affine subspace. An achievement of a series is a set of all its subsums. We study the properties of achievement sets of series whose sum range is the whole plane. It turns out that it varies on the number of Levy vectors of a series.