Correlation of fractions with divisibility constraints

Let $ B=(B-{Q}\!)-{{Q \in {\mathbb N}}} $ be an increasing sequence of positive square free integers satisfying the condition that $ B-{{Q-1}}\vert B-{{Q-2}} $ whenever Q₁ < Q₂. For any subinterval I ⊂ [0, 1], let \documentclass{article}\begin{document}$$ {\mathscr{F}-{{B}\!,-Q}(I)}=\left\lbrace a/q \in I: 1 \le a \le q \le Q, \gcd (a,q)=\gcd (q,B-{Q}\!)=1 \right\rbrace $$\end{document} It is shown that if B_Q ≪ Q^loglogQ/4, then the limiting pair correlation function of the sequence $ ({\mathscr{F}-{{B}\!,-Q}(I)})-{Q \in {\mathbb N}} $ exists and is independent of the subinterval I. Moreover, the sequence is Poissonian if $ \lim-{Q \rightarrow \infty }{{\varphi (B-{Q}\!)}\over{B-{Q}\!}} = 0 $, and exhibits a very strong repulsion if $ \lim-{Q \rightarrow \infty }{{\varphi (B-{Q}\!)}\over{B-{Q}\!}} \ne 0 $, where φ is Euler's totient function.