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.

  • $\begingroup$ I have never seen upper asymptotic density (not the symmetric one) being used for $\Bbb Z$, because it completely ignores half of the set. May I ask where have you seen it being used? $\endgroup$
    – Wojowu
    May 30, 2015 at 12:35
  • $\begingroup$ @Wojowu: I've hopefully answered your question in the OP. $\endgroup$ May 30, 2015 at 13:29
  • 1
    $\begingroup$ From my part of the world, where sequences come from invariant measures on $\{0,1\}^{\mathbb Z}$, the two quantities agree almost surely. $\endgroup$ May 30, 2015 at 17:35

2 Answers 2


Let $V$ be an affine variety defined over $\mathbb Z$, i.e., a variety defined by a (finite) set of polynomial equations in $\mathbb Z[X_1,\ldots,X_n]$. The set of integral points of $V$, denoted $V(\mathbb Z)$, is simply the set of integral solutions $\boldsymbol x =(x_1,\ldots,x_n)\in\mathbb Z^n$ to the set of equations. Also define the height of the point $\boldsymbol x$ to be $H(\boldsymbol x)=\max\{|x_1|,\ldots,|x_n|\}$. Then people who study Diophantine equations are very interested in the growth of the counting function $$ N(V(\mathbb Z),B) = \#\{\boldsymbol x\in V(\mathbb Z) : H(\boldsymbol x)\le B\}. $$ Depending on the equations defining $V$, one may get a very different result if one restricts to points with positive (or non-negative) coordinates, i.e., what one might denote $N(V(\mathbb N),B)$. From the viewpoint of arithmetic geometry, the "two-sided" counting function $N(V(\mathbb Z),B)$ is much more natural. A similar comment applies to counting projective solutions to sets of homogeneous polynomials (which more-or-less corresponds to counting points in $\mathbb P^n(\mathbb Q)$). There is a famous and much-studied conjecture of Manin et al which relates the behavior of such counting functions (and more generally over number fields) to the geometry of the variety. https://en.wikipedia.org/wiki/Manin_conjecture

  • $\begingroup$ This is far more than what I was hoping for. Thank you so much! $\endgroup$ Jun 9, 2015 at 21:55

Sorry for answering my own question, but I'd like to add a complement to Joe Silveman's answer (for those who may be interested), which however is too long to fit into a comment.

1. A reference to the "two-sided natural density" on $\mathbf Z$ is implicit to [2, Chapter 3, Section 5], where the notion is used to count the number of elements, integers, or units of a number field with height bounded above by some $b \in \mathbf R^+$.

2. I strongly agree with those arguing that a symmetric definition of the upper asymptotic density on $\mathbf Z$ is much more natural than the asymmetric definition, but I can't really mention a single paper or book on additive theory where the former is used (which may be partially due to the fact that "most" authors in this field work with densities defined on the power set of $\mathbf N^+$).

3. There is a relatively nice way to recover both definitions as a special case of a more general construction. To see how, let $\mathbf H$ denote either $\mathbf Z$, $\mathbf N$, or $\mathbf N^+$, fix $\alpha \in [-1,\infty[$, and take $\mathfrak{F} = (F_n)_{n \ge 1}$ to be a sequence of nonempty finite subsets of $\mathbf H$ such that $F_n \ne \{0\}$ for every $n$ and there exists a doubly indexed sequence $(\theta_{h,k})_{h, k \ge 1}$ of nondecreasing functions $\mathbf N^+ \to \mathbf N^+$ with the property that, for all $h, k \in \mathbf N^+$, the image of $\theta_{h,k}$ contains every sufficiently large positive integer and the following hold: $$ \lim_{n \to \infty} \frac{\sum_{i \in \Delta_n(\mathfrak{F},\theta_{h,k})} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = 0 \qquad\text{and}\qquad \lim_{n \to \infty} \frac{\sum_{i \in F_{\theta_{h,k}(n)}} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha} = \frac{1}{k^{\alpha+1}}, $$ where $\Delta_n(\mathfrak{F},\theta_{h,k})$ denotes, for each $n$, the symmetric difference of the sets $\mathbf H \cap (k^{-1} \cdot (F_n - h))$ and $ F_{\theta_{h,k}(n)}$. Next, consider the function $$ \mathsf d^\ast(\mathfrak{F};\alpha): \mathcal P(\mathbf H) \to \mathbf R: X \mapsto \limsup_{n \to \infty} \frac{\sum_{i \in X \cap F_n} |i|^\alpha}{\sum_{i \in F_n} |i|^\alpha}. $$ It is not difficult to verify that all the above conditions are satisfied, e.g., if $F_n := [\![1, n^q ]\!]$ for some $q \in \mathbf N^+$, or $\alpha \ge 0$ and $F_n := [\![ na, nb ]\!]$ for some $a,b \in \mathbf H$ with $a < b$.

This is but a definition, and not even a very appealing one. But things get perhaps more interesting if we consider the following proposition (those interested may give a look at [3, Proposition 4] for a proof):

Proposition. $\mathsf d^\ast(\mathfrak{F};\alpha)$ is an upper density.

Here an upper density (on $\mathbf H$) is a function $\mathcal P(\mathbf H) \to \mathbf R$ such that, for all $X,Y \subseteq \mathbf H$ and $h,k \in \mathbf N^+$, the following hold:

  1. $\mu^\ast(\mathbf H) = 1$;
  2. $\mu^\ast(X) \le \mu^\ast(Y)$ whenever $X \subseteq Y$;
  3. $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$;
  4. $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$.

So, based on the proposition above, and in continuity with [1, Definition 1.4], which covers the special case where $\mathbf H = \mathbf N^+$ and $F_n = [\![ 1, n ]\!]$, we may call $\mathsf d^\ast(\mathfrak{F};\alpha)$ the upper $\alpha$-density (on $\mathbf H$) relative to $\mathfrak{F}$.

In fact, $\mathsf d^\ast(\mathfrak{F};0)$ boils down to the "two-sided" (respectively, "one-sided") asymptotic density referred to in the OP in the special case where $\mathbf H = \mathbf Z$ and $F_n := [\![-n,n]\!]$ (respectively, $F_n := [\![1, n]\!]$) for all $n$.


[1] R. Giuliano-Antonini and G. Grekos, Comparison between lower and upper α-densities and lower and upper $\alpha$-analytic densities, Unif. Distrib. Theory 3 (2008), No. 2, 21-35.

[2] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.

[3] P. Leonetti and S.T., On the notions of upper and lower density, preprint (arXiv:1506.04664)


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