On maximal families of independent sets with respect to asymptotic density

TL;DR

This paper investigates the construction and properties of maximal families of subsets of natural numbers that are independent with respect to asymptotic density, revealing diverse structural and definability features.

Contribution

It introduces new methods to construct maximal density-independent families with prescribed densities and explores their complex definability and measure-theoretic properties.

Findings

Existence of maximal density-independent families with prescribed density sets.

Construction of many such families with distinct density fields.

Examples of families with pathological definability properties.

Abstract

We study families of subsets of $\omega$ 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.