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,

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We propose a continuous version of the classical Gale-Berlekamp switching game. The main results of this paper concern growth estimates for the corresponding optimization problems.