For $A\subseteq\omega$ we define the *upper density* by $$d\_u(A) = \lim\sup\_{n\to\infty}\frac{|A\cap n|}{n+1}.$$ For $y\in \omega$ we set $A - y:= \{|a\setminus y|:a\in A\}.$ Note that $|a\setminus y|$ equals the difference of $a$ and $y$ if $a\geq y$, and $0$ otherwise. The *upper Banach density* is defined by $$d\_{uB}(A) = \lim\_{n\to\infty}\big(\sup 
\{\frac{|(A-y)\cap n|}{n+1}: y\in\omega\}\big).$$
It is easy to see that $d\_{uB}(A) \geq d\_{u}(A)$ for all $A\subseteq \omega$.


**Question.** What is an example of $A\subseteq \omega$ with $d\_{uB}(A) > 0$ and $d\_u(A) = 0$?



Upper density versus upper Banach density on $\omega$
I think that it is fairly straightforward to get such a set. You can simply get the set as a union of intervals:$\newcommand{\intrvl}[2]{\langle{#1},#2)}\newcommand{\intrvr}[2]{({#1},#2\rangle}\newcommand{\limti}[1]{\lim\limits\_{{#1}\to\infty}}$
$$A=\bigcup\limits\_{n\in\omega} \intrvr{a\_n}{b\_n}.$$
And you choose the intervals in a suitable way. You want them long enough to get $d\_{uB}(A)=1$. And, at the same time, you want start each interval far enough to keep $d\_u(A)=0$. So it suffices to choose them in such way that
\begin{align\*}
\limti n (b\_n-a\_n)=\infty\\
\limti n \frac{\sum\_{k=1}^n (b\_k-a\_k)}{b\_n}=0
\end{align\*}
The first conditions will help you to get upper Banach density (a.k.a. upper uniform density) equal to one. And the second condition will keep the [asymptotic density](https://en.wikipedia.org/wiki/Natural_density) (natural density) zero.


For example, you can take $a\_n=2^n$ and $b\_n=2^n+n$.


Several posts on this site are a bit related - and the answers include some references which might be useful:


* [On the independence of lower and upper asymptotic and Banach densities](https://mathoverflow.net/q/206801)
* [Prescribed values for the uniform density](https://mathoverflow.net/q/103111)
* [Density of a set of natural numbers whose differences are not bounded.](https://mathoverflow.net/q/66191)
* [References on density of subsets of $\mathbb{N}$](https://math.stackexchange.com/q/126745) (on Mathematics Stack Exchange)


