AI 中文总结
针对循环群乘积上的傅里叶系数，证明交互阶控制的 Bohnenblust--Hille 常数具有多项式增长界，并推广至全次数类。
AI 中文摘要
固定整数 $K\ge2$，令 $C_K^N\cong(\Z/K\Z)^N$ 为 $K$ 阶循环群的乘积。对于傅里叶特征 $\chi_\alpha$，设 $s(\alpha)$ 为活跃坐标数。我们给出一个自包含的证明方案：由交互阶数控制的维数无关 Bohnenblust--Hille 常数呈多项式增长：若 $p_d=2d/(d+1)$ 且 $\gamma_2=\frac12$，$\gamma_K=\frac{K\log(K-1)}{4(K-2)}$（$K\ge3$），则 $\left(\sum_{s(\alpha)\le d} \abs{\wh f(\alpha)}^{p_d}\right)^{1/p_d} \le C_K d^{4\gamma_K+5}\norm f_\infty$。同样的估计也适用于典型的全次数类。
英文摘要
Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2πij}{K}})_{j=0}^{K-1}\}^n$ 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 the $\ell^{p_d}$ norm of Fourier coefficients $\{\hat f(α)\}$ is bounded by $L^\infty$ norm of $f$ multiplied by $C(K) d^{4γ_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5/2}$.
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