The Baire category of ideal convergent subseries and rearrangements (Q2411466)

Let \(\mathcal{I}\) be an ideal, let \(S=\{ s\in \mathbb{N}^\mathbb{N}: \forall n\in\mathbb{N} \;s(n)<s(n+1)\}\) and \(P=\{ p\in\mathbb{N}^\mathbb{N}: \;p \text{ is a bijection}\}\). These are Polish spaces with the standard Polish topology in \(\mathbb{N}^\mathbb{N}\). An ideal \(\mathcal{I}\) on \(\mathbb{N}\) is called 1-shift invariant if for any \(A\subset\mathbb{N}, A\in\mathcal{I}\) implies \(\mathbb{N} \setminus (A+1) \notin \mathcal{I}\). In the paper it is shown that if \(\mathcal{I}\) is a 1-shift invariant ideal on \(\mathbb{N}\) with the Baire property and a series \(\sum_n x_n\) with terms in a real Banach space \(X\) is not unconditionally convergent, then the sets \(\{ s\in S: \sum_n x_{s(n)} \text{ is\;} \mathcal{I}\text{-convergent}\}\) and \(\{ p\in P: \sum_n x_{p(n)} \text{ is } \mathcal{I}\text{-convergent}\}\) are meager in \(S\) and \(P\), respectively. Further, if \(\mathcal{I}\) is an ideal on \(\mathbb{N}\) with the Baire property and a series \(\sum_n x_n\) with terms in \(\mathbb{R}\) is not unconditionally convergent, then the sets \(\{ s\in S: (\sum_{i=1}^n x_{s(i)})_n \text{ is } \mathcal{I}\text{-bounded}\}\) and \(\{ p\in P: (\sum_{i=1}^n x_{p(i)})_n \text{ is } \mathcal{I}\text{-bounded}\}\) are meager in \(S\) and \(P\), respectively.