Question in the title. All of the examples I can think of (congruence classes, primes, etc.) converge as n goes to infinity.
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1$\begingroup$ you mean $\frac{1}{N} \#\{ n < N , \in E \}$ as $N \to \infty$ ? you can easily create a set for which it oscillates between to numbers $ < 1$ : $2n \in E$, and $2n+1 \in E$ iff $2^{2k} < n \le 2^{2k+1}$ $\endgroup$– reunsMay 23, 2016 at 2:16
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$\begingroup$ You're right. Is there an interesting "natural" example, rather than a set constructed explicitly to have the property? $\endgroup$– Vik78May 23, 2016 at 2:18
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$\begingroup$ you can get those kind of sets from $E = \{ n \in N \ \mid \ \sin( f(n)) > 0 \}$ for some well-chosen $f$ $\endgroup$– reunsMay 23, 2016 at 2:22
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$\begingroup$ so $E = \{n \in \mathbb{N} \ \mid \ \sin(\log_2(n)) > 0 \}$ will do the trick, its density will oscillate between $1/4$ and $3/4$ or something like that $\endgroup$– reunsMay 23, 2016 at 2:32
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