Subsums of conditionally convergent series in finite dimensional spaces
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 \cite{GM}. We obtain general results for series with harmonic-like coordinates, that is $A((-1)^{n+1}n^{-α_1},\dots,(-1)^{n+1}n^{-α_d})=\mathbb{R}^d$ for pairwise distinct numbers $α_1,\dots,α_d\in(0,1]$. For $d=2$, $α_1=1, α_2=\frac{1}{2}$ it was stated as an open problem in \cite{GM}, that is $A(\frac{(-1)^n}{n},\frac{(-1)^n}{\sqrt{n}})=\mathbb{R}^2$.
math.FA↗