I guess this is quite standard and probably easy for experts or young lovers of number theory.


For $A\subseteq\mathbb N$, denote by $d^+(A)$ its upper density, which is 
$$
d^+(A)=\lim\sup\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}
$$
Now let $A=(a\_n)$ be an increasing sequence of natural numbers and let $d\_n$ be the $n$-th difference $d\_n=a\_{n}-a\_{n-1}$.



> 
> **Question:** Suppose that $d\_n$ is not bounded, is it true that $d^+(A)=0$? 
> 
> 
> 


More generally: if $d\_n$ is not bounded, is it true that any (additive) invariant mean on $\mathbb N$ takes value $0$ over $A$?


Thanks in advance,


Valerio



Density of a set of natural numbers whose differences are not bounded.
Basic definitions:


Upper and lower asymptotic density (a.k.a. natural density):


$$\overline d(A)=\limsup \frac{A(n)}n$$


$$\underline d(A)=\liminf \frac{A(n)}n$$


Upper and lower uniform density (a.k.a. Banach density):


$$\overline u(A)=\lim\_{s\to\infty} \max\_{t\ge 0}\frac{A(t+1,t+s)}{s}$$


$$\underline u(A)=\lim\_{s\to\infty} \min\_{t\ge 0}\frac{A(t+1,t+s)}{s}$$


where $A(m,k)=|A\cap\{m,m+1,\dots,k\}|$ and $A(n)=A(1,n)$.


Note that $\underline d(\mathbb N\setminus A)=1-\overline d(A)$ and $\underline u(\mathbb N\setminus A)=1-\overline u(A)$.


It is known that $\underline u(A)\le \underline d(A) \le \overline d(A) \le \overline u(A)$, see e.g. [GLS].




---


The condition that $d\_n$ is unbounded is equivalent to $\underline u(A)=0$.


(If $d\_n$ is unbounded that we can found arbitrarily large $s$ with $A(t+1,t+s)=0$. On the other hand, if $d\_n\le M$, then $A(t+1,t+s) \ge \left\lfloor\frac{s}M\right\rfloor$ and $\underline u(A)\ge \frac 1M$.)


It is known that $\overline u(A)=\sup\{\mu(A); \mu\text{ is a shift-invariant mean on }\mathbb N\}$. The proof of this fact can be found in [B]. (EDIT: Now I realized that you have seen this result in an answer to another your question [Invariant means on the integers](https://mathoverflow.net/questions/60897/invariant-means-on-the-integers/65655#65655) Both [B] and this answer concern invariant means on $\mathbb Z$ and not $\mathbb N$, but this should not make much difference.)




---


Knowing all of this, you in fact ask whether $\underline u(A)=0$ implies $\overline d(A)=0$ or even $\overline u(A)=0$. Many counterexamples can be found. Just one of them: For the set


$$A=\mathbb N\setminus \bigcup\_{k=1}^\infty \{10^k+1,\dots,10^k+k\}$$


we get $\underline u(A)=0$ and $\underline d(A)=\overline d(A)=\overline u(A)=1$.


~~(If I remember correctly, I have seen the result that for any choice of $0\le a \le b \le c \le d\le 1$ there exists a set $A$ such that the values of $\underline u(A)$, $\underline d(A)$, $\overline d(A)$, $\overline u(A)$ are $a$, $b$, $c$ and $d$, respectively; but I might be mistaken and I cannot find any reference right now. Maybe I mixed it up with a similar result for some other type of densities.)~~


EDIT: The question whether the result mentioned in preceding paragraph holds is now posted on MO as a separate question: [On the independence of lower and upper asymptotic and Banach densities](https://mathoverflow.net/questions/206801/on-the-independence-of-lower-and-upper-asymptotic-and-banach-densities)




---


Perhaps I should mention that several equivalent definitions of Banach/uniform density appear in the literature. They are compared e.g. in [GTT]. (E.g. [B] works with a different - but equivalent - definition.)


Some references to papers where this notion was studied under the name uniform density are given in [GLS]. If I am not mistaken, the term Banach density was coined by Furstenberg [F].


[B] Mathias Beiglbock: An ultrafilter approach to Jin’s Theorem
<http://www.mat.univie.ac.at/~mathias/UltraJin_final.pdf>


[F] H. Furstenberg. Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton, 1981.


[GLS] Z. Gáliková, B. Lászlo and T. Šalát: Remarks on uniform density of sets of integers
<http://www.emis.ams.org/journals/AMI/2002/acta2002-galikova-laszlo-salat.pdf>


[GTT] Georges Grekos, Vladimír Toma and Jana Tomanová. A note on uniform or Banach density.
<http://ambp.cedram.org/ambp-bin/fitem?id=AMBP_2010__17_1_153_0>


