Polynomial Bohnenblust--Hille bounds for product of cyclic groups

Fix an integer $K\ge 2$, and let $C_K^n=\{(e^{\frac{2\pi ij}{K}}{)}_{j=0}^{K-1}{\}}^n$ be the product of cyclic groups of order $K$. For a Fourier character ${\chi }_{\alpha }$, let $s(\alpha )$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust–Hille constants governed by interaction order grow polynomially: if $p_d=2d\mathrm{/}(d+1)$ and $${\gamma }_2=\frac{1}{2},{\gamma }_K=\frac{K\log (K-1)}{4(K-2)}(K\ge 3),$$ then the ${\mathrm{\ell }}^{p_d}$ norm of Fourier coefficients $\{\widehat{f}(\alpha )\}$ is bounded by $L^{\mathrm{\infty }}$ norm of $f$ multiplied by $C(K)d^{4{\gamma }_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5\mathrm{/}2}$.