Let $\mathcal{C}$ be the collection of subsets $C$ of $\mathbb{N}$ such that
$$\lim_{m \rightarrow \infty}\frac{ \# \{k \in C\mid 1 \le k \le m \} }{m}$$ exists.
Find $A,B \in \mathcal{C}$ such that $A \cap B \notin C$.
I think I found candidates for $A,B$ but cant calculate the limit for their intersection or find an expression for their common elements; $A=\{3n\}$ and $B=\{2m+1\}$.
The classical example is to choose $A=2N$ and that $B$ contains exactly one integer from each $\{{2n,2n+1}\}$ but that this integer is $2n$ if $4^{k}⩽n<2⋅4^{k}$ for some k, and $2n+1$ if $2⋅4^{k} ⩽n<4^{k+1}$ for some k. Then the densities of A and B both exist and are both a half but the partial densities of A∩B oscillate endlessly.
By Did