Correlation of fractions with divisibility constraints

Let \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ B=(B_{Q}\!)_{{Q \in {\mathbb N}}} $\end{document} be an increasing sequence of positive square free integers satisfying the condition that $ B_{{Q_1}}\vert B_{{Q_2}} $ whenever Q 1 < Q 2 . For any subinterval I ⊂ [0, 1], let\documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{mathrsfs}\usepackage{bm} \begin{document}\pagestyle{empty}$$ {\mathscr{F}_{{B}\!,_Q}(I)}=\left\lbrace a/q \in I: 1 \le a \le q \le Q, \gcd (a,q)=\gcd (q,B_{Q}\!)=1 \right\rbrace . $$\end{document} It is shown that if B Q ≪ Q log log  Q /4 , then the limiting pair correlation function of the sequence \documentclass{article}\usepackage{amsmath,amssymb,mathrsfs,bm}\pagestyle{empty}\begin{document}$ ({\mathscr{F}_{{B}\!,_Q}(I)})_{Q \in {\mathbb N}} $\end{document} 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim