Uniform interval cancellation for binary digit parity (${\sup }_{J\subset [0,N],\theta }|{\sum }_{n\in J}(-1{)}^{s_2(n)}e(n\theta )|\ll N^{\sigma }$)

An integer interval is a disjoint union of aligned binary blocks with at most two of each length. On $[a{2}^j,(a+1)2^j)$ the Thue–Morse sequence sign factors as $f(a)f(r)$, so the Thue–Morse Fourier product bounds its exponential sum by $O(2^{\sigma j})$. Sum the geometric series over block lengths to obtain the uniform bound, for $\sigma =2-\frac{3}{4}{\log }_{2}3$.

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 $N$.