We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of order $n$ exists and $r\le\rho(n)$, where $\rho$ is the Hurwitz--Radon function. If equality fails, we obtain an explicit gap above $\sqrt n$. We also prove two asymptotic results. If $1\le m_n\le n$ and $\limsup r_n/\log_2 n<2$, there are sign forms on $\ell_2^{r_n}\times\ell_2^{m_n}\times\ell_2^n$ with norm $(1+o(1))\sqrt n$. In the square case, if $r\ge2\lceil\log_2(8n)\rceil$, then $K_{r,n}-1\ge c(1+\log_2 n)^{-4}$. We also prove a fourth-moment estimate in every fixed multilinear order, characterize equality, and give exact and asymptotic constructions.
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 $N$.
