The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:ell_{p_{1}}^{n}times cdotstimesell_{p_{m}}^{n}longrightarrowmathbb{R}$ (or $mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $left{ p_{1},...,p_{m}right} subsetlbrack2,infty]$ and when $left{ p_{1},...,p_{m}right} subsetlbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $left{ p_{1},...,p_{m}right} $ intercepts both $[2,infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
Download