Separated reciprocal-distance sum (${\sum }_j\min (Q,(2\Vert {\theta }_j\Vert {)}^{-1})\ll Q+{\delta }^{-1}\log (2Q)$)

If finitely many points on $\mathbb{R}/\mathbb{Z}$ are pairwise separated by distance modulo one at least $\delta >0$, then the displayed bound holds for $Q\ge 2$. An arc of length $2r$ contains at most $1+2r/\delta$ points. First sum over $\Vert {\theta }_j\Vert \le Q^{-1}$, then over the dyadic annuli $2^j/Q<\Vert {\theta }_j\Vert \le 2^{j+1}/Q$. Their contributions are $O(Q+{\delta }^{-1})$ and $O(Q{2}^{-j}+{\delta }^{-1})$, respectively. Summing proves the estimate. A point at zero contributes $Q$, using the convention $\min (Q,\infty )=Q$.