Hilbertian Kahane--Salem--Zygmund Inequalities: Extremizers and Quantitative Gaps

Abstract

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.