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

## TL;DR

This paper investigates the properties of achievement sets of conditionally convergent series in the plane, focusing on how these sets vary with the number of Levy vectors, building on the Levy-Steinitz theorem.

## Contribution

It extends the understanding of achievement sets for series with the entire plane as their sum range, analyzing their dependence on Levy vectors.

## Key findings

- Achievement sets vary with the number of Levy vectors.
- Sum range of these series is the entire plane.
- Properties of achievement sets are characterized based on Levy vectors.

## 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.