Ideal convergent subseries in Banach spaces

Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and ${\sum }_n{x}_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A($\mathcal{I}$):=\left{t \in {0,1}^{$\mathbb{N}$} \colon \sum_n t(n)x_n $\text{ is }$ $\mathcal{I}$$\text{-convergent}$\right}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$isnotunconditionallyconvergent,andwededucethat$A($\mathcal{I}$)$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure $\lambda$ on $\{0,1{\}}^{\mathbb{N}}$ is considered. If $\mathcal{I}$ is analytic or coanalytic, and ${\sum }_n{x}_n$ is $\mathcal{I}$-divergent, then $\lambda (A(\mathcal{I}))=0$ which extends the theorem of Dindo\v{s}, \v{S}al'at and Toma. Generalizing one of their examples, we show that, for every ideal $\mathcal{I}$ on $\mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $\lambda (A(Fin))=0$ and $\lambda (A(\mathcal{I}))=1$.