= Uniform interval cancellation for binary digit parity
{title2=$\sup_{J\subset[0,N],\theta}|\sum_{n\in J}(-1)^{s_2(n)}e(n\theta)|\ll N^\sigma$}
An integer interval is a disjoint union of aligned binary blocks with at most two of each length. On $[a2^j,(a+1)2^j)$ 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 $\sigma=2-\frac34\log_2 3$.
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