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Upper and lower densities have the strong Darboux property (1510.07473v3)

Published 26 Oct 2015 in math.CA, math.FA, and math.NT

Abstract: Let $\mathcal{P}({\bf N})$ be the power set of $\bf N$. An upper density (on $\bf N$) is a non-decreasing and subadditive function $\mu\ast: \mathcal{P}({\bf N})\to\bf R$ such that $\mu\ast({\bf N}) = 1$ and $\mu\ast(k \cdot X + h) = \frac{1}{k} \mu\ast(X)$ for all $X \subseteq \bf N$ and $h,k \in {\bf N}+$, where $k \cdot X + h := {kx + h: x \in X}$. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper P\'olya, 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 $f: \mathcal P({\bf N}) \to \bf R$ is said to have the strong Darboux property if, whenever $X \subseteq Y \subseteq \bf N$ 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 $X \subseteq \bf N$.

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