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Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \liminf_{n \to \infty} \frac{|X \cap [1, n]|}{n}$$ and $$ \mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [1, n ]|}{n},$$ and by $\mathsf{bd}_\ast(X)$ and $\mathsf{bd}^\ast(X)$, respectively, the lower and upper Banach (or uniform) density of $X$, i.e. $$\mathsf{bd}_\ast(X) := \lim_{n \to \infty} \min_{m \ge 1} \frac{|X \cap [m+1, m+n ]|}{n}$$ and $$ \mathsf{bd}^\ast(X) := \lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m+1, m+n]|}{n}.$$ It is mentioned in @Martin Sleziak's answer to Question 66191: Density of a set of natural numbers whose differences are not bounded that for all $\alpha, \beta, \gamma, \delta \in [0,1]$ with $\alpha \le \beta \le \gamma \le \delta$ there exists a set $X \subseteq \mathbf N^+$ such that $\mathsf{bd}_\ast(X) = \alpha$, $\mathsf{d}_\ast(X) = \beta$, $\mathsf{d}^\ast(X) = \gamma$, and $\mathsf{bd}^\ast(X) = \delta$.

Q. Do you know of a reference for the claim above?

Notice that Martin is not sure about the claim (he writes, "If I remember correctly, I have seen the result etc."), and from the comments to his own answer it seems he could not indeed retrieve a reference.

I am aware of results along these lines in the case where the pair $(\mathsf{d}^\ast, \mathsf{bd}^\ast)$ and its "dual" $(\mathsf{bd}_\ast, \mathsf{d}_\ast)$ are replaced, respectively, with the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ and its dual $(\mathsf{d}_\ast, \mathsf{ld}_\ast)$, where $\mathsf{ld}_\ast$ and $\mathsf{ld}^\ast$ are, respectively, the lower and upper logarithmic density on $\mathbf N^+$, namely $$\mathsf{ld}_\ast(X) := \liminf_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ and $$\mathsf{ld}^\ast(X) := \limsup_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ In fact, this can even be proven in the general case where the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ is replaced by a pair $(\mu^\ast, \nu^\ast)$ of $\mathfrak{m}$-weighted upper densities on an arithmetical semigroup $\mathbb{A} = (A, \cdot\,, |\cdot|)$ and $(\mathsf{d}_\ast, \mathsf{ld}^\ast)$ by the dual pair of $(\mu^\ast, \nu^\ast)$, under the assumption that $\mu^\ast(X) \le \nu^*(X)$ for all $X \subseteq A$ and $$\sum_{a \in A,\, |a| \le x} \mathfrak{m}(a) \sim x^s$$ for some $s > 0$ as $x \to \infty$, see

F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80

for further details. Yet, I could not find a reference for the case I am asking for.

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \liminf_{n \to \infty} \frac{|X \cap [1, n]|}{n}$$ and $$ \mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [1, n ]|}{n},$$ and by $\mathsf{bd}_\ast(X)$ and $\mathsf{bd}^\ast(X)$, respectively, the lower and upper Banach (or uniform) density of $X$, i.e. $$\mathsf{bd}_\ast(X) := \lim_{n \to \infty} \min_{m \ge 1} \frac{|X \cap [m+1, m+n ]|}{n}$$ and $$ \mathsf{bd}^\ast(X) := \lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m+1, m+n]|}{n}.$$ It is mentioned in @Martin Sleziak's answer to Question 66191: Density of a set of natural numbers whose differences are not bounded that for all $\alpha, \beta, \gamma, \delta \in [0,1]$ with $\alpha \le \beta \le \gamma \le \delta$ there exists a set $X \subseteq \mathbf N^+$ such that $\mathsf{bd}_\ast(X) = \alpha$, $\mathsf{d}_\ast(X) = \beta$, $\mathsf{d}^\ast(X) = \gamma$, and $\mathsf{bd}^\ast(X) = \delta$.

Q. Do you know of a reference for the claim above?

Notice that Martin is not sure about the claim (he writes, "If I remember correctly, I have seen the result etc."), and from the comments to his own answer it seems he could not indeed retrieve a reference.

I am aware of results along these lines in the case where the pair $(\mathsf{d}^\ast, \mathsf{bd}^\ast)$ and its "dual" $(\mathsf{bd}_\ast, \mathsf{d}_\ast)$ are replaced, respectively, with the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ and its dual $(\mathsf{d}_\ast, \mathsf{ld}_\ast)$, where $\mathsf{ld}_\ast$ and $\mathsf{ld}^\ast$ are, respectively, the lower and upper logarithmic density on $\mathbf N^+$, namely $$\mathsf{ld}_\ast(X) := \liminf_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ and $$\mathsf{ld}^\ast(X) := \limsup_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ In fact, this can even be proven in the general case where the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ is replaced by a pair $(\mu^\ast, \nu^\ast)$ of $\mathfrak{m}$-weighted upper densities on an arithmetical semigroup $\mathbb{A} = (A, \cdot\,, |\cdot|)$ and $(\mathsf{d}_\ast, \mathsf{ld}^\ast)$ by the dual pair of $(\mu^\ast, \nu^\ast)$, under the assumption that $\mu^\ast(X) \le \nu^*(X)$ for all $X \subseteq A$ and $$\sum_{a \in A,\, |a| \le x} \mathfrak{m}(a) \sim x^s$$ for some $s > 0$ as $x \to \infty$, see

F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80

for further details. Yet, I could not find a reference for the case I am asking for.

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \liminf_{n \to \infty} \frac{|X \cap [1, n]|}{n}$$ and $$ \mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [1, n ]|}{n},$$ and by $\mathsf{bd}_\ast(X)$ and $\mathsf{bd}^\ast(X)$, respectively, the lower and upper Banach (or uniform) density of $X$, i.e. $$\mathsf{bd}_\ast(X) := \lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m+1, m+n ]|}{n}$$ and $$ \mathsf{bd}^\ast(X) := \lim_{n \to \infty} \min_{m \ge 1} \frac{|X \cap [m+1, m+n]|}{n}.$$ It is mentioned in @Martin Sleziak's answer to Question 66191: Density of a set of natural numbers whose differences are not bounded that for all $\alpha, \beta, \gamma, \delta \in [0,1]$ with $\alpha \le \beta \le \gamma \le \delta$ there exists a set $X \subseteq \mathbf N^+$ such that $\mathsf{bd}_\ast(X) = \alpha$, $\mathsf{d}_\ast(X) = \beta$, $\mathsf{d}^\ast(X) = \gamma$, and $\mathsf{bd}^\ast(X) = \delta$.

Q. Do you know of a reference for the claim above?

Notice that Martin is not sure about the claim (he writes, "If I remember correctly, I have seen the result etc."), and from the comments to his own answer it seems he could not indeed retrieve a reference.

I am aware of results along these lines in the case where the pair $(\mathsf{d}^\ast, \mathsf{bd}^\ast)$ and its "dual" $(\mathsf{bd}_\ast, \mathsf{d}_\ast)$ are replaced, respectively, with the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ and its dual $(\mathsf{d}_\ast, \mathsf{ld}_\ast)$, where $\mathsf{ld}_\ast$ and $\mathsf{ld}^\ast$ are, respectively, the lower and upper logarithmic density on $\mathbf N^+$, namely $$\mathsf{ld}_\ast(X) := \liminf_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ and $$\mathsf{ld}^\ast(X) := \limsup_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ In fact, this can even be proven in the general case where the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ is replaced by a pair $(\mu^\ast, \nu^\ast)$ of $\mathfrak{m}$-weighted upper densities on an arithmetical semigroup $\mathbb{A} = (A, \cdot\,, |\cdot|)$ and $(\mathsf{d}_\ast, \mathsf{ld}^\ast)$ by the dual pair of $(\mu^\ast, \nu^\ast)$, under the assumption that $\mu^\ast(X) \le \nu^*(X)$ for all $X \subseteq A$ and $$\sum_{a \in A,\, |a| \le x} \mathfrak{m}(a) \sim x^s$$ for some $s > 0$ as $x \to \infty$, see

F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80

for further details. Yet, I could not find a reference for the case I am asking for.

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \liminf_{n \to \infty} \frac{|X \cap [1, n]|}{n}$$ and $$ \mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [1, n ]|}{n},$$ and by $\mathsf{bd}_\ast(X)$ and $\mathsf{bd}^\ast(X)$, respectively, the lower and upper Banach (or uniform) density of $X$, i.e. $$\mathsf{bd}_\ast(X) := \lim_{n \to \infty} \min_{m \ge 1} \frac{|X \cap [m+1, m+n ]|}{n}$$ and $$ \mathsf{bd}^\ast(X) := \lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m+1, m+n]|}{n}.$$ It is mentioned in @Martin Sleziak's answer to Question 66191: Density of a set of natural numbers whose differences are not bounded that for all $\alpha, \beta, \gamma, \delta \in [0,1]$ with $\alpha \le \beta \le \gamma \le \delta$ there exists a set $X \subseteq \mathbf N^+$ such that $\mathsf{bd}_\ast(X) = \alpha$, $\mathsf{d}_\ast(X) = \beta$, $\mathsf{d}^\ast(X) = \gamma$, and $\mathsf{bd}^\ast(X) = \delta$.

Q. Do you know of a reference for the claim above?

Notice that Martin is not sure about the claim (he writes, "If I remember correctly, I have seen the result etc."), and from the comments to his own answer it seems he could not indeed retrieve a reference.

I am aware of results along these lines in the case where the pair $(\mathsf{d}^\ast, \mathsf{bd}^\ast)$ and its "dual" $(\mathsf{bd}_\ast, \mathsf{d}_\ast)$ are replaced, respectively, with the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ and its dual $(\mathsf{d}_\ast, \mathsf{ld}_\ast)$, where $\mathsf{ld}_\ast$ and $\mathsf{ld}^\ast$ are, respectively, the lower and upper logarithmic density on $\mathbf N^+$, namely $$\mathsf{ld}_\ast(X) := \liminf_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ and $$\mathsf{ld}^\ast(X) := \limsup_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ In fact, this can even be proven in the general case where the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ is replaced by a pair $(\mu^\ast, \nu^\ast)$ of $\mathfrak{m}$-weighted upper densities on an arithmetical semigroup $\mathbb{A} = (A, \cdot\,, |\cdot|)$ and $(\mathsf{d}_\ast, \mathsf{ld}^\ast)$ by the dual pair of $(\mu^\ast, \nu^\ast)$, under the assumption that $\mu^\ast(X) \le \nu^*(X)$ for all $X \subseteq A$ and $$\sum_{a \in A,\, |a| \le x} \mathfrak{m}(a) \sim x^s$$ for some $s > 0$ as $x \to \infty$, see

F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80

for further details. Yet, I could not find a reference for the case I am asking for.

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \liminf_{n \to \infty} \frac{|X \cap [1, n]|}{n}$$ and $$ \mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [1, n ]|}{n},$$ and by $\mathsf{bd}_\ast(X)$ and $\mathsf{bd}^\ast(X)$, respectively, the lower and upper Banach (or uniform) density of $X$, i.e. $$\mathsf{bd}_\ast(X) := \lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m+1, m+n ]|}{n}$$ and $$ \mathsf{bd}^\ast(X) := \lim_{n \to \infty} \min_{m \ge 1} \frac{|X \cap [m+1, m+n]|}{n}.$$ It is mentioned in @Martin Sleziak's answer to Question 66191: Density of a set of natural numbers whose differences are not bounded that for all $\alpha, \beta, \gamma, \delta \in [0,1]$ with $\alpha \le \beta \le \gamma \le \delta$ there exists a set $X \subseteq \mathbf N^+$ such that $\mathsf{bd}_\ast(X) = \alpha$, $\mathsf{d}_\ast(X) = \beta$, $\mathsf{d}^\ast(X) = \gamma$, and $\mathsf{bd}^\ast(X) = \delta$.

Q. Do you know of a reference for the claim above?

Notice that Martin is not sure about the result (he writes, "If I remember correctly, I have seen the result etc."), and from the comments to his own answer it seems he could not indeed retrieve a reference.

I am aware of results along these lines in the case where the pair $(\mathsf{d}^\ast, \mathsf{bd}^\ast)$ and its "dual" $(\mathsf{bd}_\ast, \mathsf{d}_\ast)$ are replaced, respectively, with the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ and its dual, where $\mathsf{ld}_\ast$ and $\mathsf{ld}^\ast$ are, respectively, the lower and upper logarithmic density on $\mathbf N^+$, namely $$\mathsf{ld}_\ast(X) := \liminf_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ and $$\mathsf{ld}^\ast(X) := \limsup_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ In fact, this can even be proven in the general case where the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ is replaced by a pair $(\mu^\ast, \nu^\ast)$ of $\mathfrak{m}$-weighted upper densities on an arithmetical semigroup $\mathbb{A} = (A, \cdot\,, |\cdot|)$ and $(\mathsf{d}_\ast, \mathsf{ld}^\ast)$ by the dual pair of $(\mu^\ast, \nu^\ast)$, under the assumption that $\mu^\ast(X) \le \nu^*(X)$ for all $X \subseteq A$ and $$\sum_{a \in A,\, |a| \le x} \mathfrak{m}(a) \sim x^s$$ for some $s > 0$ as $x \to \infty$, see

F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80

for further details. Yet, I could not find a reference for the case I am asking for.

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \liminf_{n \to \infty} \frac{|X \cap [1, n]|}{n}$$ and $$ \mathsf{d}^\ast(X) := \limsup_{n \to \infty} \frac{|X \cap [1, n ]|}{n},$$ and by $\mathsf{bd}_\ast(X)$ and $\mathsf{bd}^\ast(X)$, respectively, the lower and upper Banach (or uniform) density of $X$, i.e. $$\mathsf{bd}_\ast(X) := \lim_{n \to \infty} \max_{m \ge 1} \frac{|X \cap [m+1, m+n ]|}{n}$$ and $$ \mathsf{bd}^\ast(X) := \lim_{n \to \infty} \min_{m \ge 1} \frac{|X \cap [m+1, m+n]|}{n}.$$ It is mentioned in @Martin Sleziak's answer to Question 66191: Density of a set of natural numbers whose differences are not bounded that for all $\alpha, \beta, \gamma, \delta \in [0,1]$ with $\alpha \le \beta \le \gamma \le \delta$ there exists a set $X \subseteq \mathbf N^+$ such that $\mathsf{bd}_\ast(X) = \alpha$, $\mathsf{d}_\ast(X) = \beta$, $\mathsf{d}^\ast(X) = \gamma$, and $\mathsf{bd}^\ast(X) = \delta$.

Q. Do you know of a reference for the claim above?

Notice that Martin is not sure about the claim (he writes, "If I remember correctly, I have seen the result etc."), and from the comments to his own answer it seems he could not indeed retrieve a reference.

I am aware of results along these lines in the case where the pair $(\mathsf{d}^\ast, \mathsf{bd}^\ast)$ and its "dual" $(\mathsf{bd}_\ast, \mathsf{d}_\ast)$ are replaced, respectively, with the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ and its dual $(\mathsf{d}_\ast, \mathsf{ld}_\ast)$, where $\mathsf{ld}_\ast$ and $\mathsf{ld}^\ast$ are, respectively, the lower and upper logarithmic density on $\mathbf N^+$, namely $$\mathsf{ld}_\ast(X) := \liminf_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ and $$\mathsf{ld}^\ast(X) := \limsup_{n \to \infty} \frac{\sum_{x \in X \cap [1,n]} \frac{1}{x}}{\log n}.$$ In fact, this can even be proven in the general case where the pair $(\mathsf{ld}^\ast, \mathsf{d}^\ast)$ is replaced by a pair $(\mu^\ast, \nu^\ast)$ of $\mathfrak{m}$-weighted upper densities on an arithmetical semigroup $\mathbb{A} = (A, \cdot\,, |\cdot|)$ and $(\mathsf{d}_\ast, \mathsf{ld}^\ast)$ by the dual pair of $(\mu^\ast, \nu^\ast)$, under the assumption that $\mu^\ast(X) \le \nu^*(X)$ for all $X \subseteq A$ and $$\sum_{a \in A,\, |a| \le x} \mathfrak{m}(a) \sim x^s$$ for some $s > 0$ as $x \to \infty$, see

F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80

for further details. Yet, I could not find a reference for the case I am asking for.