Let $A \subset \mathbb{N}$ have $\overline{d}(A):= \limsup_{N\to \infty} \frac{|A\cap \{1,\ldots,N\}|}{N} >2/3$. Is it true that $\overline{d}(A \cap A/2)>0$?
What I can show is that this holds for sets $A$ with upper density at least $3/4$, but my method does not generalize for sets of smaller density. On the other hand, any example I can think of satisfies this property.
Any ideas would be appreciated.
Note: Here $A/2 := \{n\in \mathbb{N}: 2n \in A\}$.