= Small Type I sums for binary digit parity
{title2=$N^{-1}\sum_{m\le N^{1/100}}|\sum_{n\in I_m}(-1)^{s_2(mn)}|\ll N^{\sigma+1/100-1}$}
The additive <character orthogonality> filter $1_{m\mid r}=m^{-1}\sum_{a=0}^{m-1}e(ar/m)$ restricts the <uniform interval cancellation for binary digit parity> to multiples of $m$, without increasing its bound. Each inner <Type I sum> is therefore $O(N^\sigma)$. Summing the at most $N^{1/100}$ outer terms proves the displayed estimate, which is smaller than every fixed negative power of $\log N$ for large $N$.
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