On unimodular multilinear forms with small norms on sequence spaces

Abstract

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 $A_{m,n}:{\ell }_{p_1}^n\times \cdots \times {\ell }_{p_m}^n⟶\mathbb{R}$ (or $\mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left\{p_1,...,p_m\right\}\subset [2,\infty ]$ and when $\left\{p_1,...,p_m\right\}\subset [1,2)$ 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.

7 pages