Ideal convergent subseries in Banach spaces

TL;DR

This paper investigates the size and category of the set of subseries in Banach spaces that are convergent with respect to an ideal, revealing conditions under which this set is meager or measure-zero, and providing examples with contrasting properties.

Contribution

It extends existing results by analyzing the measure and category of ideal-convergent subseries in Banach spaces, including new examples with contrasting measure properties.

Findings

If the ideal has the Baire property and the series is not unconditionally convergent, the set is meager.

For analytic or coanalytic ideals, the measure of the set is zero when the series is ideal-divergent.

Existence of divergent series with measure-one sets for some ideals and measure-zero for others.

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 $A(\mathcal{I}):=\left\{t \in \{0,1\}^{\mathbb{N}} \colon \sum_n t(n)x_n \textrm{ is } \mathcal{I}\textrm{-convergent}\right\}$. 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 $\lambda$ 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 $\lambda(A(\mathcal{I}))=1$.