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

$$A_{\varepsilon} := \{z \in \mathbb N \ | \ \text {The set of } n \in \mathbb N \text { such that } nz \in S\text { has upper density at least } \varepsilon\}$$

has nonzero lower density?

Remark: This question arose when trying to prove a number theory result involving greatest common denominators of sets of naturals.