The upper asymptotic density on $\mathbf Z$, viz. the function $$ {\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n}, $$ has a ''symmetric variant'', which is given by the function $$ {\sf d}_{\rm ev}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [-n,n]|}{2n+1} $$ and is, in some way, better suited to the integers, insofar as ${\sf d}^\ast(X) = {\sf d}^\ast(X \cap \mathbf N^+)$ for every $X \subseteq \mathbf Z$. Yet, I'm sadly unaware of any place in the literature where the use of ${\sf d}_{\rm ev}^\ast$ instead of ${\sf d}^\ast$ makes a substantial difference. So the question is:

Question. Do you have any reference to suggest in this respect?

Of course, the same question can be asked for other ''upper densities'' (most notably, the upper Banach density and the upper logarithmic density), as well as for ''lower densities'' and ''plain densities''. And I would be happy also with information on these.

Edit 3 (June 9, 2015). I emailed @ValerioCapraro, who answered that he doesn't recall where he had picked up the definitions of the Beurling upper and lower density mentioned in his question. However, he suggested to give a look at the collected works of Beurling on harmonic analysis, and following his suggestion I've found that Beurling has the following notion: Given a uniformly discrete set $X$ of real numbers, define the uniform upper density of $X$ by $${\sf u.u.d.}(X) := \lim_{s \to \infty} \sup_{t \in \mathbf R} \frac{|X \cap [t,t+s]|}{s}.$$ (The limit exists by Fekete's lemma.) See e.g. Section I in:

A. Beurling, "Local harmonic analysis with some applications to differential operators", pp. 109-125 in: A. Gelbard (ed.), Some Recent Advances in the Basic Sciences, Vol. I, Belfer Grad. School of Science, Annual Science Conference Proc., Academic Press, 1966. Reprinted in: The Collected Works of Arne Beurling, Vol.2: Harmonic Analysis, Birkhäuser, 1989 (pp. 299-315).

This generalizes the Banach upper density to $\mathbf R$, and is half of an answer to the (main) question I'm posing, but not quite an answer.

Edit 2 (June 9, 2015). I found a reference to ${\sf d}_{\rm ev}^\ast$ in a relatively old thread, where @ValerioCapraro writes of the upper (and lower) Beurling density on the integers, which is exactly what I'm looking for. This may have something to do with amenable groups, but I'm not familiar with the subject, and couldn't find an explicit reference to these items in a paper, book, or similia. I mean, there is a lot of literature on "Beurling densities", but I'm hoping for something focused, on the one hand, on the case in which I'm essentially interested (the integers) and explaining, on the other, why people could, should, or might be interested in having a symmetric definition along with the asymmetric one. Any hint?

Edit 1 (May 30, 2015). To answer a question by @Wojowu in the comments below, the definition of $\mathsf{d}^\ast$ that I'm considering here is the one used, e.g., by Halberstam and Roth in Sequences (Springer, 1983), where the authors write (p. xvii):

The 'integer sequences' under investigation will usually be subsequences of the sequence of non-negative integers. But from time to fime it will be convenient to consider more general integer sequences; for example, in [...] Chapter I we will admit (monotone strictly increasing integer) sequences containing negative integers, whilst not all the sequences considered in Chapter IV are monotone.

Notice also the footnote on the same page:

One reason for not 'counting' the non-positive elements, even if there may be any finite number of these, will be explained in the text.

The upper asymptotic density on $\mathbf Z$, viz. the function $$ {\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n}, $$ has a ''symmetric variant'', which is given by the function $$ {\sf d}_{\rm ev}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [-n,n]|}{2n+1} $$ and is, in some way, better suited to the integers, insofar as ${\sf d}^\ast(X) = {\sf d}^\ast(X \cap \mathbf N^+)$ for every $X \subseteq \mathbf Z$. Yet, I'm sadly unaware of any place in the literature where the use of ${\sf d}_{\rm ev}^\ast$ instead of ${\sf d}^\ast$ makes a substantial difference. So the question is:

Question. Do you have any reference to suggest in this respect?

Of course, the same question can be asked for other ''upper densities'' (most notably, the upper Banach density and the upper logarithmic density), as well as for ''lower densities'' and ''plain densities''. And I would be happy also with information on these.

Edit 3 (June 9, 2015). I emailed @ValerioCapraro, who answered that he doesn't recall where he had picked up the definitions of the Beurling upper and lower density mentioned in his question. However, he suggested to give a look at the collected works of Beurling on harmonic analysis, and following his suggestion I've found that Beurling has the following notion: Given a uniformly discrete set $X$ of real numbers, define the uniform upper density of $X$ by $${\sf u.u.d.}(X) := \lim_{s \to \infty} \sup_{t \in \mathbf R} \frac{|X \cap [t,t+s]|}{s}.$$ (The limit exists by Fekete's lemma.) See e.g. Section I in:

A. Beurling, "Local harmonic analysis with some applications to differential operators", pp. 109-125 in: A. Gelbard (ed.), Some Recent Advances in the Basic Sciences, Vol. I, Belfer Grad. School of Science, Annual Science Conference Proc., Academic Press, 1966. Reprinted in: The Collected Works of Arne Beurling, Vol.2: Harmonic Analysis, Birkhäuser, 1989 (pp. 299-315).

This generalizes the Banach upper density to $\mathbf R$, and is half of an answer to the (main) question I'm posing, but not quite an answer.

Edit 2 (June 9, 2015). I found a reference to ${\sf d}_{\rm ev}^\ast$ in a relatively old thread, where @ValerioCapraro writes of the upper (and lower) Beurling density on the integers, which is exactly what I'm looking for. This may have something to do with amenable groups, but I'm not familiar with the subject, and couldn't find an explicit reference to these items in a paper, book, or similia. I mean, there is a lot of literature on "Beurling densities", but I'm hoping for something focused, on the one hand, on the case in which I'm essentially interested (the integers) and explaining, on the other, why people could, should, or might be interested in having a symmetric definition along with the asymmetric one. Any hint?

Edit 1 (May 30, 2015). To answer a question by @Wojowu in the comments below, the definition of $\mathsf{d}^\ast$ that I'm considering here is the one used, e.g., by Halberstam and Roth in Sequences (Springer, 1983), where the authors write (p. xvii):

The 'integer sequences' under investigation will usually be subsequences of the sequence of non-negative integers. But from time to fime it will be convenient to consider more general integer sequences; for example, in [...] Chapter I we will admit (monotone strictly increasing integer) sequences containing negative integers, whilst not all the sequences considered in Chapter IV are monotone.

Notice also the footnote on the same page:

One reason for not 'counting' the non-positive elements, even if there may be any finite number of these, will be explained in the text.

The upper asymptotic density on $\mathbf Z$, viz. the function $$ {\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n}, $$ has a ''symmetric variant'', which is given by the function $$ {\sf d}_{\rm ev}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [-n,n]|}{2n+1} $$ and is, in some way, better suited to the integers, insofar as ${\sf d}^\ast(X) = {\sf d}^\ast(X \cap \mathbf N^+)$ for every $X \subseteq \mathbf Z$. Yet, I'm sadly unaware of any place in the literature where the use of ${\sf d}_{\rm ev}^\ast$ instead of ${\sf d}^\ast$ makes a substantial difference. So the question is:

Question. Do you have any reference to suggest in this respect?

Of course, the same question can be asked for other ''upper densities'' (most notably, the upper Banach density and the upper logarithmic density), as well as for ''lower densities'' and ''plain densities''. And I would be happy also with information on these.

Edit 3 (June 9, 2015). I emailed @ValerioCapraro, who answered that he doesn't recall where he had picked up the definitions of the Beurling upper and lower density mentioned in his question. However, he suggested to give a look at the collected works of Beurling on harmonic analysis, and following his suggestion I've found that Beurling has the following notion: Given a uniformly discrete set $X$ of real numbers, define the uniform upper density of $X$ by $${\sf u.u.d.}(X) := \lim_{s \to \infty} \sup_{t \in \mathbf R} \frac{X \cap [s,s+t]}{t}.$$ (The limit exists by Fekete's lemma.) See e.g. Section I in:

A. Beurling, "Local harmonic analysis with some applications to differential operators", pp. 109-125 in: A. Gelbard (ed.), Some Recent Advances in the Basic Sciences, Vol. I, Belfer Grad. School of Science, Annual Science Conference Proc., Academic Press, 1966. Reprinted in: The Collected Works of Arne Beurling, Vol.2: Harmonic Analysis, Birkhäuser, 1989 (pp. 299-315).

This generalizes the Banach upper density to $\mathbf R$, and is half of an answer to the (main) question I'm posing, but not quite an answer.

Edit 2 (June 9, 2015). I found a reference to ${\sf d}_{\rm ev}^\ast$ in a relatively old thread, where @ValerioCapraro writes of the upper (and lower) Beurling density on the integers, which is exactly what I'm looking for. This may have something to do with amenable groups, but I'm not familiar with the subject, and couldn't find an explicit reference to these items in a paper, book, or similia. I mean, there is a lot of literature on "Beurling densities", but I'm hoping for something focused, on the one hand, on the case in which I'm essentially interested (the integers) and explaining, on the other, why people could, should, or might be interested in having a symmetric definition along with the asymmetric one. Any hint?

Edit 1 (May 30, 2015). To answer a question by @Wojowu in the comments below, the definition of $\mathsf{d}^\ast$ that I'm considering here is the one used, e.g., by Halberstam and Roth in Sequences (Springer, 1983), where the authors write (p. xvii):

The 'integer sequences' under investigation will usually be subsequences of the sequence of non-negative integers. But from time to fime it will be convenient to consider more general integer sequences; for example, in [...] Chapter I we will admit (monotone strictly increasing integer) sequences containing negative integers, whilst not all the sequences considered in Chapter IV are monotone.

Notice also the footnote on the same page:

One reason for not 'counting' the non-positive elements, even if there may be any finite number of these, will be explained in the text.

The upper asymptotic density on $\mathbf Z$, viz. the function $$ {\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n}, $$ has a ''symmetric variant'', which is given by the function $$ {\sf d}_{\rm ev}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [-n,n]|}{2n+1} $$ and is, in some way, better suited to the integers, insofar as ${\sf d}^\ast(X) = {\sf d}^\ast(X \cap \mathbf N^+)$ for every $X \subseteq \mathbf Z$. Yet, I'm sadly unaware of any place in the literature where the use of ${\sf d}_{\rm ev}^\ast$ instead of ${\sf d}^\ast$ makes a substantial difference. So the question is:

Question. Do you have any reference to suggest in this respect?

Of course, the same question can be asked for other ''upper densities'' (most notably, the upper Banach density and the upper logarithmic density), as well as for ''lower densities'' and ''plain densities''. And I would be happy also with information on these.

Edit 3 (June 9, 2015). I emailed @ValerioCapraro, who answered that he doesn't recall where he had picked up the definitions of the Beurling upper and lower density mentioned in his question. However, he suggested to give a look at the collected works of Beurling on harmonic analysis, and following his suggestion I've found that Beurling has the following notion: Given a uniformly discrete set $X$ of real numbers, define the uniform upper density of $X$ by $${\sf u.u.d.}(X) := \lim_{s \to \infty} \sup_{t \in \mathbf R} \frac{|X \cap [t,t+s]|}{s}.$$ (The limit exists by Fekete's lemma.) See e.g. Section I in:

A. Beurling, "Local harmonic analysis with some applications to differential operators", pp. 109-125 in: A. Gelbard (ed.), Some Recent Advances in the Basic Sciences, Vol. I, Belfer Grad. School of Science, Annual Science Conference Proc., Academic Press, 1966. Reprinted in: The Collected Works of Arne Beurling, Vol.2: Harmonic Analysis, Birkhäuser, 1989 (pp. 299-315).

This generalizes the Banach upper density to $\mathbf R$, and is half of an answer to the (main) question I'm posing, but not quite an answer.

Edit 2 (June 9, 2015). I found a reference to ${\sf d}_{\rm ev}^\ast$ in a relatively old thread, where @ValerioCapraro writes of the upper (and lower) Beurling density on the integers, which is exactly what I'm looking for. This may have something to do with amenable groups, but I'm not familiar with the subject, and couldn't find an explicit reference to these items in a paper, book, or similia. I mean, there is a lot of literature on "Beurling densities", but I'm hoping for something focused, on the one hand, on the case in which I'm essentially interested (the integers) and explaining, on the other, why people could, should, or might be interested in having a symmetric definition along with the asymmetric one. Any hint?

Edit 1 (May 30, 2015). To answer a question by @Wojowu in the comments below, the definition of $\mathsf{d}^\ast$ that I'm considering here is the one used, e.g., by Halberstam and Roth in Sequences (Springer, 1983), where the authors write (p. xvii):

The 'integer sequences' under investigation will usually be subsequences of the sequence of non-negative integers. But from time to fime it will be convenient to consider more general integer sequences; for example, in [...] Chapter I we will admit (monotone strictly increasing integer) sequences containing negative integers, whilst not all the sequences considered in Chapter IV are monotone.

Notice also the footnote on the same page:

One reason for not 'counting' the non-positive elements, even if there may be any finite number of these, will be explained in the text.

Do you have any reference to suggest in this respect?

Edit (May 30, 2015). To answer a question by @Wojowu in the comments below, the definition of $\mathsf{d}^\ast$ that I'm considering here is the one used, e.g., by Halberstam and Roth in Sequences (Springer, 1983), where the authors write (p. xvii):

The 'integer sequences' under investigation will usually be subsequences of the sequence of non-negative integers. But from time to fime it will be convenient to consider more general integer sequences; for example, in [...] Chapter I we will admit (monotone strictly increasing integer) sequences containing negative integers, whilst not all the sequences considered in Chapter IV are monotone.

Notice also the footnote on the same page:

One reason for not 'counting' the non-positive elements, even if there may be any finite number of these, will be explained in the text.

Edit 2 (June 9, 2015). I found a reference to ${\sf d}_{\rm ev}^\ast$ in a relatively old question, where @ValerioCapraro makes reference to the upper (and lower) Beurling density on the integers, which is exactly what I'm looking for. This may have something to do with amenable groups, but I'm not familiar with the subject, and couldn't find an explicit reference to these items in a paper, book, or similia. I mean, there is a lot of literature on "Beurling densities", but I'm hoping for something focused, on the one hand, on the case in which I'm essentially interested (the integers) and explaining, on the other, why on earth people could be interested in having a symmetric definition along with the asymmetric one (which I can't see). Any hint?

Question. Do you have any reference to suggest in this respect?

Edit 3 (June 9, 2015). I emailed @ValerioCapraro, who answered that he doesn't recall where he had picked up the definitions of the Beurling upper and lower density mentioned in his question. However, he suggested to give a look at the collected works of Beurling on harmonic analysis, and following his suggestion I've found that Beurling has the following notion: Given a uniformly discrete set $X$ of real numbers, define the uniform upper density of $X$ by $${\sf u.u.d.}(X) := \lim_{s \to \infty} \sup_{t \in \mathbf R} \frac{X \cap [s,s+t]}{t}.$$ (The limit exists by Fekete's lemma.) See e.g. Section I in:

A. Beurling, "Local harmonic analysis with some applications to differential operators", pp. 109-125 in: A. Gelbard (ed.), Some Recent Advances in the Basic Sciences, Vol. I, Belfer Grad. School of Science, Annual Science Conference Proc., Academic Press, 1966. Reprinted in: The Collected Works of Arne Beurling, Vol.2: Harmonic Analysis, Birkhäuser, 1989 (pp. 299-315).

This generalizes the Banach upper density to $\mathbf R$, and is half of an answer to the (main) question I'm posing, but not quite an answer.

Edit 2 (June 9, 2015). I found a reference to ${\sf d}_{\rm ev}^\ast$ in a relatively old thread, where @ValerioCapraro writes of the upper (and lower) Beurling density on the integers, which is exactly what I'm looking for. This may have something to do with amenable groups, but I'm not familiar with the subject, and couldn't find an explicit reference to these items in a paper, book, or similia. I mean, there is a lot of literature on "Beurling densities", but I'm hoping for something focused, on the one hand, on the case in which I'm essentially interested (the integers) and explaining, on the other, why people could, should, or might be interested in having a symmetric definition along with the asymmetric one. Any hint?

Edit 1 (May 30, 2015). To answer a question by @Wojowu in the comments below, the definition of $\mathsf{d}^\ast$ that I'm considering here is the one used, e.g., by Halberstam and Roth in Sequences (Springer, 1983), where the authors write (p. xvii):

The 'integer sequences' under investigation will usually be subsequences of the sequence of non-negative integers. But from time to fime it will be convenient to consider more general integer sequences; for example, in [...] Chapter I we will admit (monotone strictly increasing integer) sequences containing negative integers, whilst not all the sequences considered in Chapter IV are monotone.

Notice also the footnote on the same page:

One reason for not 'counting' the non-positive elements, even if there may be any finite number of these, will be explained in the text.