Thue–Morse Fourier product
= Thue–Morse Fourier product
{c}
{title2=$\sum_{r<2^j}(-1)^{s_2(r)}e(r\theta)=\prod_{l=0}^{j-1}(1-e(2^l\theta))$}
Each <binary digit> independently contributes either $1$ or $-e(2^l\theta)$, proving the product. Two adjacent factors have modulus $8|\cos(\pi t)|(1-\cos^2(\pi t))\le16/(3\sqrt3)<4$. Consequently the product is $O(2^{\sigma j})$, with $\sigma=2-\frac34\log_2 3<1$. This two-factor estimate suffices for a power saving, though its exponent is not optimal.