 Thue–Morse Fourier product

Each binary digit independently contributes either $1$ or $-e(2^l\theta )$, proving the product. Two adjacent factors have modulus $8|\cos (\pi t)|(1-{\cos }^2(\pi t))\le 16/(3\sqrt{3})<4$. Consequently the product is $O(2^{\sigma j})$, with $\sigma =2-\frac{3}{4}{\log }_{2}3<1$. This two-factor estimate suffices for a power saving, though its exponent is not optimal.