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 Am,n : ℓ n p 1 ×· · ·×ℓ n pm −→ R (or C) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of Am,n when {p1, ..., pm} ⊂ [2, ∞] and when {p1, ..., pm} ⊂ [1, 2) are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: {p1, ..., pm} intercepts both [2, ∞] 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.2010 Mathematics Subject Classification. 15A60, 15B05,

Constants of the Kahane--Salem--Zygmund inequality asymptotically bounded by $1$ II

“…However, in many instances, deterministic arguments are not available and probabilistic methods come into play. In the 1970's and 1980's several authors have followed this vein (see [4,11,15,16,21]) and nowadays these probabilistic approaches are explored in different lines of research (see [2,3,6,7,8,10,12,13,14,17,19] and the references therein). In general, the probabilistic arguments are enough to provide optimal exponents but the constants involved are not precise.…”