Properties of simple density ideals

TL;DR

This paper introduces and studies simple density ideals, a class of ideals based on upper density functions, revealing their vast diversity, maximality in Kat907 order, and the uniqueness of generating functions.

Contribution

It defines simple density ideals, proves the existence of many non-isomorphic and incomparable examples, and analyzes the uniqueness of functions generating the same ideal.

Findings

There are 4c4 many non-isomorphic simple density ideals.

A large family of pairwise incomparable simple density ideals can be constructed.

The ideal 4Z4 of sets with asymptotic density zero is maximal in Kat907 order among simple density ideals.

Abstract

Let $G$ consist of all functions $g \colon \omega \to [0,\infty)$ with $g(n) \to \infty$ and $\frac{n}{g(n)} \nrightarrow 0$. Then for each $g\in G$ the family $\mathcal{Z}_g=\{A\subseteq\omega:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal associated to the notion of so-called upper density of weight $g$. Although those ideals have recently been extensively studied, they do not have their own name. In this paper, for Reader's convenience, we propose to call them simple density ideals. We show that there are $\mathfrak{c}$ many non-isomorphic (in fact even incomparable with respect to Kat\v{e}tov order) simple density ideals. Moreover, we prove that for a given $A\subset G$ with $\text{card}(A)<\mathfrak{b}$ one can construct a family of cardinality $\mathfrak{c}$ of pairwise incomparable (with respect to inclusion) simple density ideals which additionally are incomparable with all $\mathcal{Z}_g$ for $g\in A$. We show that this cannot be generalized to Kat\v{e}tov order as the ideal $\mathcal{Z}$ of sets of asymptotic density zero is maximal in the sense of Kat\v{e}tov order among all simple density ideals. We examine how many substantially different functions $g$ can generate the same ideal $\mathcal{Z}_g$ -- it turns out that the answer is either $1$ or $\mathfrak{c}$ (depending on $g$).