Note: By upper/lower density, we shall mean the upper/lower asymptotic density as given here.
Question:
For any subset $S \subset \mathbb N$ with positive upper density, does there exists a $\varepsilon > 0$ such that the set
$$A_{\varepsilon} := \{z \in \mathbb N \ | \ \text {The set } S_z=\{n~|~nz \in S\}\text { has upper density at least } \varepsilon\}$$
has nonzero lower density?
Remarks:
This question arose when trying to prove a number theory result involving greatest common denominators of sets of naturals.
The corresponding result, if one replaces the set $A_\varepsilon$ with the set $B := \{z \in \mathbb N \ | \ nz \in S \text { for infinitely many } n \in \mathbb N \}$ is true, and not overly difficult to prove. In fact we even have an explicit bound for the lower density of $B$ - it is at least the upper density of $S$.
