On maldistributed sequences and meager ideals

TL;DR

This paper characterizes meager ideals on natural numbers via the concept of maldistribution of sequences in Polish spaces, linking topological properties with measure-theoretic and submeasure conditions.

Contribution

It establishes a new equivalence between meager ideals and maldistribution conditions involving Polish space sequences and submeasures, extending previous results.

Findings

Meager ideals are characterized by maldistribution of sequences in Polish spaces.

The equivalence involves a specific submeasure u and a technical condition from prior work.

An analogue involving lower semicontinuous submeasures is also proven.

Abstract

We show that an ideal $\mathcal{I}$ on $\omega$ is meager if and only if the set of sequences $(x_n)$ taking values in a Polish space $X$ for which all elements of $X$ are $\mathcal{I}$-cluster points of $(x_n)$ is comeager. The latter condition is also known as $\nu$-maldistribution, where $\nu: \mathcal{P}(\omega)\to \mathbb{R}$ is the $\{0,1\}$-valued submeasure defined by $\nu(A)=1$ if and only if $A\notin \mathcal{I}$. It turns out that the meagerness of $\mathcal{I}$ is also equivalent to a technical condition given by Misik and Toth in [J. Math. Anal. Appl. 541 (2025), 128667]. Lastly, we show that the analogue of the first part holds replacing $\nu$ with $\|\cdot\|_\varphi$, where $\varphi$ is a lower semicontinuous submeasure.