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arXiv · 2603.28922

On maximal families of independent sets with respect to asymptotic density

Jonathan M. Keith·Paolo Leonetti
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Abstract

We study families of subsets of $ω$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that $\mathsf{d}[\mathcal{A}]$ attains a prescribed set of values in $(0,1)$ with at most countably many exceptions. In addition, under $\mathrm{cov}(\mathcal{N})=\mathfrak{c}$, it is possible to construct such $\mathcal{A}$ with no exceptions. We also construct $2^{\mathfrak{c}}$ maximal $\mathsf{d}$-independent families with pairwise distinct generated density fields and obtain maximal families with strong definability pathologies, including examples without the Baire property and, consistently, nonmeasurable examples.

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Jonathan M. Keith, Paolo Leonetti. 2026-03-30. On maximal families of independent sets with respect to asymptotic density. https://arxiv.org/abs/2603.28922

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