 4:40 AM
Jan 24 '15 at 18:58, by Martin Sleziak
Maybe it will be useful to remind that instructions how to use MathJax in SE chat can be found here and here.
I believe that some approach to asymptotic density for subsets of $\mathbb N\times\mathbb N$ can be found in literature.
The ones I have seen use the ratio $A(m,n)/mn$ where $A(m,n)=|\{(i,j)\in A; i\le n, j\le n\}$.
There is this paper:
John Christopher: The Asymptotic Density of Some k-Dimensional Sets. The American Mathematical Monthly, Vol. 63, No. 6 (Jun. - Jul., 1956), pp. 399-401. jstor.org/stable/2309400
Google scholar provides texts citing this article and related articles.
However, in this paper the author uses only diagonal values, i.e. $m=n$. To me this seems less natural.
I have seen some other approaches using some kind of convergence of double sequences. If I remember correctly, it was Pringsheim convergence.
For example these two.
F. Moricz: Statistical convergence of multiple sequences, Archiv der Mathematik, 2003. dx.doi.org/10.1007/s00013-003-0506-9
Again, I will add Google Scholar links for citing articles and related articles.
Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, Journal of Math. Analysis and application, 288(2003), 223-231. doi.org/10.1016/j.jmaa.2003.08.004

1 hour later… 5:50 AM
Just in case somebody stumbles upon the above messages, it was in context of this:
@Arbuja I am not sure to which extent this might be useful for you, but I have collected a few references to asymptotic density for subsets of $\mathbb N\times\mathbb N$ here. — Martin Sleziak 1 hour ago