On unimodular multilinear forms with small norms on sequence spaces

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 $A_{m,n}:\ell_{p_{1}}^{n}\times \cdots\times\ell_{p_{m}}^{n}\longrightarrow$\mathbb{R}$$(or$$\mathbb{C}$$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left{ p_{1},…,p_{m}\right} \subset\lbrack2,\infty]$andwhen$\left{ p_{1},…,p_{m}\right} \subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $\{p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_m\}$ intercepts both $[2,\mathrm{\infty }]$ 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.