Gaps between numbers of the form $pq$

Question

Mathematicians keep improving Zhang's bound on gaps between primes. According to Wikipedia, there are infinitely many pairs of primes such that their difference is no more than 246. This is all very exciting.

What does it tell us about gaps between products of two distinct primes? Can we say that there are infinitely many pairs of numbers of the form $pq$, for distinct primes $p$ and $q$, such that their difference is less than some $K$? To be clear, I mean pairs of numbers $(a,b)$, where $a=p_1q_1$ and $b=p_2q_2$, where $p_i\neq q_i$, but it's okay if $p_i=q_j$, or $p_i=p_j$ when $i\neq j$.

Thanks in advance for any insights on this.

Answer 1

Goldston, Graham, Pintz, and Yildirim showed that one can take $K=6$ in your statement! (another link) Indeed they could show this even before Zhang's breakthrough.