= Upper asymptotic density
{title2=$\overline d(A)$}
= Upper natural density
{synonym}
For a <subset> $A$ of the <positive integers>, its upper asymptotic density is
$$
\overline d(A)=\limsup_{N\to\infty}\frac{|A\cap\{1,\ldots,N\}|}{N}.
$$
The <limit superior> keeps the largest limiting proportion even when the running proportions oscillate. Because they lie in $[0,1]$, this <limit superior> always exists in $[0,1]$. If <natural density> exists, <upper asymptotic density> equals it.
The convention here uses $1,\ldots,N$, while <natural density> above uses $0,\ldots,N-1$. Replacing one interval by the other changes a count by at most one, so the proportions differ by at most $1/N$ and have the same <limit superior>. For the <set without natural density> above, the count in $\{1,\ldots,16\}$ is six rather than five, because $16$ is included and zero is absent. Nevertheless its <upper asymptotic density> is exactly $2/3$. The even <positive integers> have <upper asymptotic density> $1/2$ and the <prime numbers> have <upper asymptotic density> zero.
The useful feature is that a large proportion along arbitrarily large initial <integer intervals> already supplies structure. The <Szemerédi theorem> says that positive <upper asymptotic density> forces <arithmetic progressions> of every finite length, with positive common difference, even without an existing <natural density>. Szemerédi's 1975 paper resolved this general finite-length problem; it makes density a bridge from counting to additive structure. https://doi.org/10.4064/aa-27-1-199-245[Endre Szemerédi, On sets of integers containing k elements in arithmetic progression, Acta Arithmetica 27, 199–245, 1975]{ref}
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