Changes of signs in conditionally convergent series on a small set (Q654195)

An ideal \(I\) is called \textit{proper} if \(\mathbb{N}\) does not belong to \(I\) and it is called \textit{admissible} if any finite subset of \(\mathbb{N}\) is in \(I\). In this paper the authors consider only ideals that are proper and admissible. \textit{R. Filipów} and \textit{P. Szuca} [J. Math. Anal. Appl. 362, No. 1, 64--71 (2010; Zbl 1188.40001)] introduced the \((R)\) property which is defined as follows. The ideal \(I\) has the \((R)\) property if for any conditionally convergent series of real numbers \(\sum_{n\in\mathbb{N}}a_n\) and \(s\in\mathbb{\overline{R}}:=\mathbb{R}\cup\{-\infty,\infty\}\), there is a permutation \(\pi\) of \(\mathbb{N}\) such that \(\sum_{n\in\mathbb{N}}a_{\pi(n)}=s\) and \(\{n\in\mathbb{N}:\pi(n)\neq n\}\in I\). The authors define the \((S)\) property: \(I\) has the \((S)\) property if given conditionally convergent series of real numbers \(\sum_{n\in\mathbb{N}}a_n\) and \(s\in\mathbb{\overline{R}}\), then there is a sequence of signs \(\delta=(\delta_n)_{n\in\mathbb{N}}\) such that \(\sum_{n\in\mathbb{N}}\delta_n a_n=s\) and \(N(\delta):=\{n\in\mathbb{N}: \delta_n=-1\}\in I\). They obtain the following result: If the series \(\sum_{n\in\mathbb{N}}a_n\) of real numbers is conditionally convergent and \(s\in\mathbb{\overline{R}}\), then there exists a sequence of signs \((\delta_n)_{n\in\mathbb{N}}\) such that \(\sum_{n\in\mathbb{N}}\delta_n a_n=s\). The main result in this paper is that the \((R)\) property is equivalent to the \((S)\) property.