Density of a difference set of a set wih zero upper Banach density The upper Banach density of a subset $A\subset\mathbb N$ is defined
$$d^*(A)=\limsup_{M-N\to\infty}\frac{|A\cap[M,M+1,\cdots,N]|}{M-N}$$
also the the upper density and lower density of a set $A$ is defined as following respectively
$$\bar{d}(A)=\limsup_{n\to\infty}\frac{|A\cap[1,\cdots,n]|}{n}$$
$$\underline{d}(A)=\liminf_{n\to\infty}\frac{|A\cap[1,\cdots,n]|}{n}.$$
It is known that if $d^\ast(A)\gt 0$, then $A-A$ is $\Delta^*$. It means that $A-A$ has non-empty intersection with difference set of any sequence.
Now I want to know that Is there any subset $A\subset\mathbb N$ such that $\bar{d}(A-A)=1$?
 A: Answering to the question formulated in the comments ("Is there a set $P$ with upper density $1$ such that $A-A$ is not contained in $P$ for any set $A$ with positive upper Banach density?"), note that for any $A$ with $\bar d(A)>0$, if $r>1/\bar d(A)$ then for any integers $a_1,...,a_r$ the total density of the sets $A-a_1,A-a_2,...,A-a_r$ is greater than $1$. Therefore $(A-a_i)\cap(A-a_j)$ is non-empty for some pair $(i,j)$.
In other words, $A-A$ intersects the set of differences $\Delta(a_1,...,a_r)=\{a_i-a_j:1\leq i<j\leq r\}$.
Now construct $P$ as to avoid difference sets of arbitrarily large sets, but still have upper density $1$, for instance:
$$P:=\mathbb{N}\setminus\left[\bigcup_{r=1}^\infty10^r\Delta(1,2,...,r)\right]$$
(here I used the usual notation, for $n\in\mathbb{N}$ and $X\subset\mathbb{N}$ denote by $nX:=\left\{nx:x\in X\right\}$).
Thus $P$ has upper density $1$ (and actually Banach lower density $1$), but for any $A$ with positive Banach upper density $A-A\not\subset P$.
