= Separated reciprocal-distance sum
{title2=$\sum_j\min(Q,(2\|\theta_j\|)^{-1})\ll Q+\delta^{-1}\log(2Q)$}
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 $Q\ge2$. An arc of length $2r$ contains at most $1+2r/\delta$ points. First sum over $\|\theta_j\|\le Q^{-1}$, then over the dyadic annuli $2^j/Q<\|\theta_j\|\le2^{j+1}/Q$. Their contributions are $O(Q+\delta^{-1})$ and $O(Q2^{-j}+\delta^{-1})$, respectively. Summing proves the estimate. A point at zero contributes $Q$, using the convention $\min(Q,\infty)=Q$.
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