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Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an (arithmetic) upper density. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found here and here.

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by $$\mu_\star(X)=1-\mu^\star(X^c).$$ Now, all examples I have in mind are superadditive functions, namely $$ \mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y) $$ whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:

Question. Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then $$ 1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)? $$

In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive.

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an (arithmetic) upper density. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found here and here.

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by $$\mu_\star(X)=1-\mu^\star(X^c).$$ Now, all examples I have in mind are superadditive functions, namely $$ \mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y) $$ whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:

Question. Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then $$ 1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)? $$

In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive.

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an upper density. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found here and here.

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by $$\mu_\star(X)=1-\mu^\star(X^c).$$ Now, all examples I have in mind are superadditive functions, namely $$ \mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y) $$ whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:

Question. Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then $$ 1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)? $$

In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive.

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an (arithmetic) upper density. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found here and here.

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by $$\mu_\star(X)=1-\mu^\star(X^c).$$ Now, all examples I have in mind are superadditive functions, namely $$ \mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y) $$ whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:

Question. Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then $$ 1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)? $$

In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive.

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an upper density. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found here and here.

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by $$\mu_\star(X)=1-\mu^\star(X^c).$$ Now, all examples I have in mind are superadditive functions, namely $$ \mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y) $$ whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:

Question. Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then $$ 1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)? $$

In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive, and its domain is closed under finite unions.

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:

(F1) $\mu^\star(\mathbf{N}^+)=1$;

(F2) $\mu^\star(X) \le \mu^\star(Y)$ if $X\subseteq Y$;

(F3) $\mu^\star(X\cup Y) \le \mu^\star(X)+\mu^\star(Y)$;

(F4) $\mu^\star(\{kx:x \in X\})=\mu^\star(X)/k$ for all $k \in \mathbf{N}^+$;

(F5) $\mu^\star(\{x+h: x \in X\})=\mu^\star(X)$ for all $h \in \mathbf{N}^+$.

A function of this type is said to be an upper density. The set of these functions include the upper asymptotic, Banach, logarithmic, analytic, Polya, Buck densities and many others. Related questions on these type of functions can be found here and here.

At this point we can defined its associated lower density $\mu_\star$ for all $X\subseteq \mathbf{N}^+$ by $$\mu_\star(X)=1-\mu^\star(X^c).$$ Now, all examples I have in mind are superadditive functions, namely $$ \mu_\star(X\cup Y) \ge \mu_\star(X)+\mu_\star(Y) $$ whenever $X$ and $Y$ are disjoint subsets of $\mathbf{N}^+$. Does this property hold in general?

After an easy manipulation, the question turns out to be equivalent to the following:

Question. Let $\mu^\star$ be an upper density on $\mathbf{N}^+$, that is, a function satisfying axioms (F1)-(F5). Is it true that if $X,Y$ are subsets of $\mathbf{N}^+$ such that $X\cup Y=\mathbf{N}^+$ then $$ 1+\mu^\star(X\cap Y) \le \mu^\star(X)+\mu^\star(Y)? $$

In turn, this can be viewed as a strenghtening of (F3) above; moreover, it would imply that the induced density $\mu$, which can be seen as the restriction of $\mu^\star$ on $\{X\subseteq \mathbf{N}^+:\mu^\star(X)=\mu_\star(X)\}$, is additive.