Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 3 / b

Write $\alpha =a/q+\beta$, where $|\beta |\le q^{-2}$. Split the $m$-interval into $O(M/q+1)$ consecutive blocks of length at most $q/2$. If distinct integers $m,n$ lie in one block, then $0<|m-n|<q/2$. Since $(a,q)=1$,whereas $|\beta (m-n)|\le 1/(2q)$. Hence $\Vert \alpha m-\alpha n\Vert \ge 1/(2q)$.

The points $\alpha m$ in one block are therefore $1/(2q)$-separated modulo one. Order them by distance from the nearest integer. Apart from a bounded number of endpoints, the $j$th closest point has distance $\gg j/q$, and consequentlyMultiplying by $O(M/q+1)$ givesThis is the reciprocal fractional-part sum near a rational estimate.