On a Diophantine inequality with reciprocals

A sharpened lower bound is obtained for the number of solutions to an inequality of the form $\alpha \le \{(a\overline {n}+bn)/q\}<\beta $, $1\le n\le N$, where $q$ is a sufficiently large prime number, $a$ and $b$ are integers with $(ab,q)=1$, $n\overline {n}\equiv 1 \pmod q$, and $0\le \alpha <\beta \le 1$. The length $N$ of the range of the variable $n$ is of order $q^\varepsilon $, where $\varepsilon >0$ is an arbitrarily small fixed number.