On maldistributed sequences and meager ideals

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.