Geometric properties of points on modular hyperbolas

Abstract:

Given an integer $n\ge 2$, let $\mathcal {H}_n$ be the set \[ \mathcal {H}_n= \{(a,b) \ : \ ab \equiv 1 \pmod n,\ 1\le a,b \le n-1\} \] and let $M(n)$ be the maximal difference of $b-a$ for $(a,b) \in \mathcal {H}_n$. We prove that for almost all $n$, $n-M(n)=O\left (n^{1/2+o(1)}\right ).$ We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of $\mathcal {H}_n$.