Achievement sets of conditionally convergent series

TL;DR

This paper explores the complex structure of achievement sets of conditionally convergent series in multiple dimensions, providing new examples and theoretical insights into their topological properties.

Contribution

It introduces novel examples and general theorems that reveal the complexity of achievement sets in higher dimensions for conditionally convergent series.

Findings

Multidimensional achievement sets are more complex than in the real line case.

Presented many surprising examples of achievement sets.

Developed general theorems capturing underlying ideas.

Abstract

Considering the sets of subsums of series (or achievement sets) we show that for conditionally convergent series the multidimensional case is much more complicated than that of the real line. Although we are far from the full topological classification of such sets, we present many surprising examples and catch the ideas standing behind them in general theorems.