# Can anyone give an example of a set of numbers with arithmetic density that doesn't converge to a limit?

Question in the title. All of the examples I can think of (congruence classes, primes, etc.) converge as n goes to infinity.

• 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 $2^{2k} < n \le 2^{2k+1}$ – reuns May 23 '16 at 2:16
• You're right. Is there an interesting "natural" example, rather than a set constructed explicitly to have the property? – Vik78 May 23 '16 at 2:18
• you can get those kind of sets from $E = \{ n \in N \ \mid \ \sin( f(n)) > 0 \}$ for some well-chosen $f$ – reuns May 23 '16 at 2:22
• 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 – reuns May 23 '16 at 2:32