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 $\mathrm{limsup}{r}_n/{\log }_{2}n<2$, 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 $r\ge 2\lceil {\log }_2(8n)\rceil$, 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.