AI 中文总结
该研究针对有限循环群$C_q^N$上的函数，证实其无维度Bohnenblust-Hille常数随次数次指数增长，推导了相关最优常数的次指数上界，并应用得到傅里叶层Bohr半径的双侧估计及渐近行为。
AI 中文摘要
设$C_q$表示q次单位根群，Becker、Klein、Slote、Volberg和Zhang的工作提出一个问题：$C_q^N$上函数的无维度Bohnenblust-Hille常数是否随次数次指数增长。我们肯定回答了该问题。事实上，对傅里叶特征最多涉及d个坐标的函数，我们证明了更强的估计：若$\text{BH}_{d,q}$是该更大类函数的最优常数，则对每个固定的$q\text{≥}2$，$\text{BH}_{d,q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right)$，其中$c_2=2$，$q\text{≥}3$时$c_q=\sqrt{2q\log(q-1)/(q-2)}$。作为应用，我们得到了恰好涉及d个坐标的特征形成的傅里叶层的Bohr半径的双侧估计，并确定了其在d与N的自然联合区域中的渐近行为。
英文摘要
Let $C_q$ denote the group of the $q$th roots of unity. A question arising from the work of Becker, Klein, Slote, Volberg and Zhang is whether the dimension-free Bohnenblust--Hille constants for functions on $C_q^N$ grow subexponentially with the degree. We answer this question affirmatively. In fact, we prove a stronger estimate for functions whose Fourier characters involve at most $d$ coordinates. If $\BHint{d}{q}$ is the optimal constant for this larger class, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. As an application, we obtain two-sided estimates for the Bohr radius of the Fourier layer formed by characters involving exactly $d$ coordinates, and we determine its asymptotic behaviour in natural joint regimes of $d$ and $N$.
CommentsRevised and substantially improved version. The exposition and organization have been strengthened, and several proofs have been expanded for clarity. The main results are unchanged
