4:40 AM
Jan 24 '15 at 18:58, by Martin Sleziak
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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
However, in this paper the author uses only diagonal values, i.e. $m=n$. To me this seems less natural.
@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