On maldistributed sequences and meager ideals
Paolo Leonetti
TL;DR
The paper addresses when an ideal $\mathcal{I}$ on $\omega$ is meager by connecting it to maldistribution properties of sequences in Polish spaces. It introduces $\nu$-maldistribution with $\nu=\mathbf{1}_{\mathcal{I}^+}$ and shows that $\mathcal{I}$ is meager iff the set of $\nu$-maldistributed sequences, $\Sigma_\nu(X)$, is comeager for any nontrivial Polish $X$, equivalently when a Mišík–Tóth-type condition holds. It then generalizes to diffuse submeasures obtained from lower semicontinuous submeasures $\varphi$, proving comeagerness of $\Sigma_{\|\cdot\|_\varphi}(X)$ under the corresponding hypotheses and describing analytic $P$-ideals as $\mathrm{Exh}(\varphi)$. The results provide a unifying view linking ideal topology with dynamical-like distribution properties of sequences, and supply a broad family of examples and constructions for analytic $P$-ideals and their associated maldistribution phenomena.
Abstract
We show that an ideal $\mathcal{I}$ on $ω$ 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 $ν$-maldistribution, where $ν: \mathcal{P}(ω)\to \mathbb{R}$ is the $\{0,1\}$-valued submeasure defined by $ν(A)=1$ 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 $ν$ with $\|\cdot\|_\varphi$, where $\varphi$ is a lower semicontinuous submeasure.