This is an announcement for the paper “Ideal convergent subseries in Banach spaces” by Marek Balcerzakhttps://arxiv.org/find/math/1/au:+Balcerzak_M/0/1/0/all/0/1, Michał Popławskihttps://arxiv.org/find/math/1/au:+Poplawski_M/0/1/0/all/0/1, Artur Wachowiczhttps://arxiv.org/find/math/1/au:+Wachowicz_A/0/1/0/all/0/1.

Abstract: 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 $A(\mathcal{I}):={t\in{0,1}^{\mathbb{N}}: \sum_n t(n)x_n\textrm{is} \mathcal{I}-\textrm{convergent}}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure λ on {0,1}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 $\lambda(A(\mathcal{I}))=1$.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.03699