The Green-Tao theorem states that if $A$ is an infinite subset of the the prime numbers such that
$\limsup_{n \to \infty} \frac{ |A \cap [1, n]| }{\pi(n)} > 0$ then for any integer $k$, $A$ contains an arithmetic progression of length $k$.
My question is the following: Suppose that $A = \{p_{n_k}\}$ such that $\sum_{n_k} \frac{1}{p_{n_k}} = \infty$. Does this set contain arbitrarily long arithmetic progressions?