Hausdorff-Young inequalities for $\mathrm{S}\mathrm{U}(1,1)$ Fourier products indexed by binary cubes

We study nonlinear logarithmic Hausdorff-Young inequalities with constant $1$ for $\mathrm{S}\mathrm{U}(1,1)$-valued Fourier products indexed by binary cubes. The inequalities are proved for the optimal range of exponents, extending into the region $1\mathrm{/}p+1\mathrm{/}q<1$. The proof combines majorization, a known sharp two-point inequality, and tree iteration. We also obtain the corresponding chain inequality and nonlinear bounds for generating functions of alternating chains, recovering the known sharp bounds for additive energies on the binary cube as the amplitudes vanish.