Let $S \subset \mathbb N$ be a subset. The natural density is defined as
$$D(S) = \lim_{n \to \infty} \frac{|E \cap \{1, \cdots, n\}|}{n}$$
whenever this limit exists.
So question is the following: Let $S \subset \mathbb N$ be a set of natural density zero. Let $E = \{n_k : k\in \mathbb N\}$ be a subsequence of $\mathbb N$. Assume that $E$ has positive upper density, that is
$$ D^*(E) = \limsup_{n\to \infty} \frac{|E \cap \{1, \cdots, n\}|}{n}>0$$
Does the set $E\cap S$ has natural density zero in $E$? That is, do we have
$$\lim_{k \to \infty} \frac{|\{k :n_k \in S\}\cap \{1, \cdots, k\}|}{k} = 0 \ \ ?$$
This question is related to the following:
a simple question about the density convergence of sequences
(Also not sure if the "Number theory" tag is suitable, feel free to edit)
