On the notions of upper and lower density

Abstract

Let $\mathcal{P}({\bf N})$ be the power set of ${\bf N}$. We say that a function $μ^\ast: \mathcal{P}({\bf N}) \to \bf R$ is an upper density if, for all $X,Y\subseteq{\bf N}$ and $h, k\in{\bf N}^+$, the following hold: (F1) $μ^\ast({\bf N}) = 1$; (F2) $μ^\ast(X) \le μ^\ast(Y)$ if $X \subseteq Y$; (F3) $μ^\ast(X \cup Y) \le μ^\ast(X) + μ^\ast(Y)$; (F4) $μ^\ast(k\cdot X) = \frac{1}{k} μ^\ast(X)$, where $k \cdot X:=\{kx: x \in X\}$; (F5) $μ^\ast(X + h) = μ^\ast(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $α$-densities (with $α$ 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 $μ^\ast(X)\le 1$ 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.