On the notions of upper and lower density

TL;DR

This paper formalizes the concept of upper densities on the natural numbers, unifies various classical densities under a common framework, and explores their properties and independence of axioms.

Contribution

It introduces a general definition of upper densities, shows many classical densities fit this framework, and analyzes the independence of axioms and their properties.

Findings

Classical upper densities satisfy the new axiomatic framework.

The axioms (F1)-(F5) are mutually independent.

Replacing (F2) with a weaker condition extends the theory.

Abstract

Let $\mathcal{P}({\bf N})$ be the power set of ${\bf N}$. We say that a function $\mu^\ast: \mathcal{P}({\bf N}) \to \bf R$ is an upper density if, for all $X,Y\subseteq{\bf N}$ and $h, k\in{\bf N}^+$, the following hold: (F1) $\mu^\ast({\bf N}) = 1$; (F2) $\mu^\ast(X) \le \mu^\ast(Y)$ if $X \subseteq Y$; (F3) $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$; (F4) $\mu^\ast(k\cdot X) = \frac{1}{k} \mu^\ast(X)$, where $k \cdot X:=\{kx: x \in X\}$; (F5) $\mu^\ast(X + h) = \mu^\ast(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $\alpha$-densities (with $\alpha$ a real parameter $\ge -1$), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (F1)-(F5), and we investigate various properties of upper densities (and related functions) under the assumption that (F2) is replaced by the weaker condition that $\mu^\ast(X)\le 1$ for every $X\subseteq{\bf N}$. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.