Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups

Abstract

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 $N$.