Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density $d^+(A)=\lim\sup_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$. It seems to me I've found somewhere the following result, but I am not able to find the reference and I am not completely sure that I remember well:
for any $r$ in the interval $[d^{-}(A),d^+(A)]$, there is an invariant mean $m$ on $l^\infty(\mathbb Z)$ such that $m(\chi_A)=r$. Moreover, if $d^+(A)=d^{-}(A)$, then every invariant mean gives $\chi_A$ the same value and this is just the density $d(A)$.