The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function $$ \mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\mathsf{d}^\star(A), $$ where $\mathscr{A}$ stands for the set of finite unions of arithmetic progressions of $\mathbf{N}^+$ and $\mathsf{d}^\star$ is the classical asymptotic upper density $$ \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ Something on the upper Buck density can be seen, for example, here. Related MO questions concerning the set of upper densities can be found in T1, T2, or T3.

Since $\mathfrak{b}^\star$ is an upper density in this sense, it could be natural to ask whether one can obtain a representation like the one of $\mathsf{d}^\star$; more precisely:

Question. Does there exist a sequence of real-valued functions $(f_n)_{n\ge 1}$ such that each $f_n$ is defined on the power set of $\{1,\ldots,n\}$ and for each $X\subseteq \mathbf{N}^+$ it holds $$ \mathfrak{b}^\star(X)=\limsup_{n\to \infty}f_n(X \cap \{1,\ldots,n\})\,\,\,\,? $$

The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function $$ \mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\mathsf{d}^\star(A), $$ where $\mathscr{A}$ stands for the set of finite unions of arithmetic progressions of $\mathbf{N}^+$ and $\mathsf{d}^\star$ is the classical asymptotic upper density $$ \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ Something on the upper Buck density can be seen, for example, here. Related MO questions concerning the set of upper densities can be found in T1, T2, or T3.

Since $\mathfrak{b}^\star$ is an upper density in this sense, it could be natural to ask whether one can obtain a representation like the one of $\mathsf{d}^\star$; more precisely:

Question. Does there exist a sequence of real-valued functions $(f_n)_{n\ge 1}$ such that each $f_n$ is defined on the power set of $\{1,\ldots,n\}$ and for each $X\subseteq \mathbf{N}^+$ it holds $$ \mathfrak{b}^\star(X)=\limsup_{n\to \infty}f_n(X \cap \{1,\ldots,n\})\,\,\,\,? $$

The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function $$ \mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\mathsf{d}^\star(A), $$ where $\mathscr{A}$ stands for the set of finite unions of arithmetic progressions of $\mathbf{N}^+$ and $\mathsf{d}^\star$ is the classical asymptotic upper density $$ \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ The upper Buck density has been studied in literature, see e.g. here or here. Related MO questions concerning the set of upper densities can be found in T1, T2, or T3.

Since $\mathfrak{b}^\star$ is an upper density in this sense, it could be natural to ask whether one can obtain a representation like the one of $\mathsf{d}^\star$; more precisely:

Question. Does there exist a sequence of real-valued functions $(f_n)_{n\ge 1}$ such that each $f_n$ is defined on the power set of $\{1,\ldots,n\}$ and for each $X\subseteq \mathbf{N}^+$ it holds $$ \mathfrak{b}^\star(X)=\limsup_{n\to \infty}f_n(X \cap \{1,\ldots,n\})\,\,\,\,? $$

The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function $$ \mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\mathsf{d}^\star(A), $$ where $\mathscr{A}$ stands for the set of finite unions of arithmetic progressions of $\mathbf{N}^+$ and $\mathsf{d}^\star$ is the classical asymptotic upper density $$ \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\mapsto \limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ Something on the upper Buck density can be seen, for example, here. Related MO questions concerning the set of upper densities can be found in T1, T2, or T3.

Since $\mathfrak{b}^\star$ is an upper density in this sense, it could be natural to ask whether one can obtain a representation like the one of $\mathsf{d}^\star$; more precisely:

Question. Does there exist a sequence of real-valued functions $(f_n)_{n\ge 1}$ such that each $f_n$ is defined on the power set of $\{1,\ldots,n\}$ and for each $X\subseteq \mathbf{N}^+$ it holds $$ \mathfrak{b}^\star(X)=\limsup_{n\to \infty}f_n(X \cap \{1,\ldots,n\})\,\,\,\,? $$