Another characterization of meager ideals

TL;DR

This paper characterizes meager ideals on positive integers through the existence of a bounded real sequence with specific subsequence and permutation properties related to $ ext{I}$-limit points and cluster points, providing a new perspective on their structure.

Contribution

It offers a novel characterization of meager ideals using bounded sequences and the topological size of sets of subsequences and permutations preserving $ ext{I}$-limit points.

Findings

Characterization of meager ideals via bounded sequences and $ ext{I}$-limit points.

Existence of sequences where subsequences preserving $ ext{I}$-limit points form a comeager set.

Analogous results for $ ext{I}$-cluster points.

Abstract

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of $\mathcal{I}$-limit points is comeager and, in addition, every accumulation point of $x$ is also an $\mathcal{I}$-limit point (that is, a limit of a subsequence $(x_{n_k})$ such that $\{n_1,n_2,\ldots,\} \notin \mathcal{I}$). The analogous characterization holds also for $\mathcal{I}$-cluster points.