Polynomial Bohnenblust--Hille Bounds for product of cyclic groups

Source abstract

Fix an integer $K\ge2$, and let $C_K^N\cong(\Z/K\Z)^N$ be the product of cyclic groups of order $K$. For a Fourier character $χ_α$, let $s(α)$ 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 $p_d=2d/(d+1)$ and $$γ_2=\frac12, \qquad γ_K=\frac{K\log(K-1)}{4(K-2)}\quad(K\ge3),$$ then \[ \left(\sum_{s(α)\le d} \abs{\wh f(α)}^{p_d}\right)^{1/p_d} \le C_K d^{4γ_K+5}\norm f_\infty. \] The same estimate follows for the canonical total-degree class.