Reciprocal fractional-part sum near a rational

If $(a,q)=1$, $|\alpha -a/q|\le q^{-2}$, and $M,R\ge 2$, then uniformly in $m_0$,Split the interval into blocks of length at most $q/2$. Within a block the points $\alpha m$ are $\gg 1/q$-separated modulo one; ordering their distances from the nearest integer gives $O(R+q\log q)$ per block.