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Let $\mathsf{d}^\star$ be the asymptotic upper density on $\mathbf{N}$, that is, for each $X\subseteq \mathbf{N}$ we have $\mathsf{d}^\star(X)=\limsup_n |X\cap [1,n]|/n$.

Then, is it possible to find a set $A$ of even numbers and a set $B$ of odd numbers such that $$ \mathsf{d}^\star(A\cup B) \neq \mathsf{d}^\star(A)+\mathsf{d}^\star(B)? $$

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The set $C$ of natural numbers whose binary representation has an odd length has upper asymptotic density $\frac{2}{3}$. The same happens for the set $D$ of natural numbers whose binary representation has an even length. In particular $d^\star(C)+d^\star(D)=\frac{4}{3}$ while $d^\star(C\cup D)=1$.

So, if we take $A=2C$ and $B=2D+1$, we have: $$d^\star(A)=d^\star(B)=\frac{1}{3},\qquad d^\star(A\cup B)=\frac{1}{2}.$$

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    $\begingroup$ ...caro Jack :P $\endgroup$ – Paolo Leonetti Jul 30 '15 at 23:30

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