On maldistributed sequences and meager ideals

Abstract

We show that an ideal $\mathcal{I}$ on $\ddot{I}‰$ 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 $\hat{I}½$-maldistribution, where $\hat{I}½:\mathcal{P}(\ddot{I}‰)\to \mathbb{R}$ is the $\{0,1\}$-valued submeasure defined by $\hat{I}½(A)=1$ 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 $\hat{I}½$ with $\Vert \cdot {\Vert }_{\ddot{I}}†$, where $\ddot{I}†$ is a lower semicontinuous submeasure.

J. Math. Anal. Appl., to appear