Let $A\subseteq\mathbb N$, as usual we set $d^+(A)=\lim\sup\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$ and 
$d^{-}=\lim\inf\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$. It's very stardard that in general $d^+(A)\neq d^-(A)$. My question is little different: is there any $A$ for which $d^-(A)=0$ and $d^+(A)\neq0$? Equal to $1$?


Thanks in advance,
Valerio



A set with lower density equal to $0$ and upper density different from $0$
For an explicit example, you could use $A= \{ n: \lfloor \log\_2 \log\_2{n} \rfloor \text{ is even} \} $.


