4:40 AM
I believe that some approach to asymptotic density for subsets of $\mathbb N\times\mathbb N$ can be found in literature.
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
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.
F. Moricz: Statistical convergence of multiple sequences, Archiv der Mathematik, 2003. dx.doi.org/10.1007/s00013-003-0506-9
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
@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
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