Subsums of conditionally convergent series in finite dimensional spaces

TL;DR

This paper investigates the achievement sets of conditionally convergent series in finite-dimensional spaces, providing new results that answer open problems about the structure of these sets for specific harmonic-like series.

Contribution

It offers general results on achievement sets for series with harmonic-like coordinates, solving open problems in the field.

Findings

Achievement sets of certain series are the entire space .

Confirmed open problem for series with /n and / in D.

Extended understanding of achievement sets for conditionally convergent series.

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 \cite{GM}. 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 \cite{GM}, that is $A(\frac{(-1)^n}{n},\frac{(-1)^n}{\sqrt{n}})=\mathbb{R}^2$.