Subsums of conditionally convergent series in finite dimensional spaces

Abstract

An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of conditionally convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems formulated in [S. G\l \c{a}b, J. Marchwicki, Levy--Steinitz theorem and achievement sets of conditionally convergent series on the real plane, J. Math. Anal. Appl. 459 (2018) 476--489]. We obtain general results for series with harmonic-like coordinates, that is $A((-1)^{n+1}n^{-\alpha_1},\dots,(-1)^{n+1}n^{-\alpha_d})=\mathbb{R}^d$ for pairwise distinct numbers $\alpha_1,\dots,\alpha_d\in(0,1]$. For $d=2$, $\alpha_1=1, \alpha_2=\frac{1}{2}$ it was stated as an open problem in [S. G\l \c{a}b, J. Marchwicki, Levy--Steinitz theorem and achievement sets of conditionally convergent series on the real plane, J. Math. Anal. Appl. 459 (2018) 476--489], that is $A(\frac{(-1)^n}{n},\frac{(-1)^n}{\sqrt{n}})=\mathbb{R}^2$.