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 $\bar{d}(A-A)=1$?