Small Type I sums for binary digit parity ($N^{-1}{\sum }_{m\le N^{1/100}}|{\sum }_{n\in I_m}(-1{)}^{s_2(mn)}|\ll N^{\sigma +1/100-1}$)

Let $f(r)=(-1{)}^{s_2(r)}$ be the Thue–Morse sequence, let $e(t)=e^{2\pi it}$, and let $N\ge 2$. For each positive integer $m\le N^{1/100}$, take an integer interval $I_m\subseteq [1,N/m]$. The range condition ensures that every product $mn$ remains at most $N$.

Choose the integer interval $J_m$ from $m\min I_m$ to $m\max I_m$, omitting an empty $I_m$. Character orthogonality givesThe uniform interval cancellation for binary digit parity bounds each inner exponential sum by $C{N}^{\sigma }$, with $\sigma =2-\frac{3}{4}{\log }_{2}3<1$. The average over $a$ does not increase that bound. ThereforeThe exponent is negative, so this normalized Type I sum bound is smaller than every fixed negative power of $\log N$ for large $N$. No cancellation between the outer terms is needed; each inner progression is controlled separately.