Given an integer
n
≥
2
n\ge 2
, let
H
n
\mathcal {H}_n
be the set
\[
H
n
=
{
(
a
,
b
)
:
a
b
≡
1
(
mod
n
)
,
1
≤
a
,
b
≤
n
−
1
}
\mathcal {H}_n= \{(a,b) \ : \ ab \equiv 1 \pmod n,\ 1\le a,b \le n-1\}
\]
and let
M
(
n
)
M(n)
be the maximal difference of
b
−
a
b-a
for
(
a
,
b
)
∈
H
n
(a,b) \in \mathcal {H}_n
. We prove that for almost all
n
n
,
n
−
M
(
n
)
=
O
(
n
1
/
2
+
o
(
1
)
)
.
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
H
n
\mathcal {H}_n
.