= Solution
Use the distinct reduced fractions $a/q$ with $1\le q\le Q$, $0\le a<q$ and $(a,q)=1$, regarded in $\mathbb R/\mathbb Z$. The fraction zero appears as $0/1$; one is the same circle point and is not added a second time. For two distinct such points, the ordinary difference and its possible wrapped complement are nonzero <integer> multiples of $1/(qq')$. Hence
$$
\boxed{\|a/q-a'/q'\|_{\mathbb R/\mathbb Z}\ge1/(qq')\ge Q^{-2}.}
$$
This proves the required separation of the <Farey fractions>.
Back to article page
