DISTRIBUTION OF GAPS BETWEEN THE INVERSES $\mathrm{mod} q

Let $q$ be a positive integer, let $\mathcal{I}=\mathcal{I}(q)$ and $\mathcal{J}=\mathcal{J}(q)$ be subintervals of integers in $[1,q]$ and let $\mathcal{M}$ be the set of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approaches infinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution between consecutive elements of $\mathcal{M}$ becomes exponential. AMS 2000 Mathematics subject classification: Primary 11K06; 11B05; 11N69