Upper and lower densities have the strong Darboux property

Abstract

Let $\mathcal{P}(\mathbf{N})$ be the power set of $\mathbf{N}$. An upper density (on $\mathbf{N}$) is a non\-decreasing and subadditive function ${\mu }^{\ast }:\mathcal{P}(\mathbf{N})\to \mathbf{R}$ such that ${\mu }^{\ast }(\mathbf{N})=1$ and ${\mu }^{\ast }(k\cdot X+h)=\frac{1}{k}{\mu }^{\ast }(X)$ for all $X\subseteq \mathbf{N}$ and $h,k\in {\mathbf{N}}^{+}$, where $k\cdot X+h:=\{kx+h:x\in X\}$. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper Pólya, and upper analytic densities are examples of upper densities. We show that every upper density ${\mu }^{\ast }$ has the strong Darboux property, and so does the associated lower density, where a function $f:\mathcal{P}(\mathbf{N})\to \mathbf{R}$ is said to have the strong Darboux property if, whenever $X\subseteq Y\subseteq \mathbf{N}$ and $a\in [f(X),f(Y)]$, there is a set $A$ such that $X\subseteq A\subseteq Y$ and $f(A)=a$. In fact, we prove the above under the assumption that the monotonicity of ${\mu }^{\ast }$ is relaxed to the weaker condition that ${\mu }^{\ast }(X)\le 1$ for every $X\subseteq \mathbf{N}$.

10 pages, no figures. Fixed minor details and streamlined the exposition. To appear in Journal of Number Theory

Cited by in corpus (5)

• On the notions of upper and lower density
• On the density of sumsets
• On the density of sumsets, II
• Abstract densities and ideals of sets
• On Small Sets of Integers