Sobre normas de formas unimodulares em espaços de sequências

Abstract

The study of multilinear forms A: ℓnp 1 ×· · ·×ℓnp m → K with coefficients ±1 (or matrices with ±1 entries) has several applications in various branches of mathematics and has been investigated by several authors in different contexts since the end of the 19th century. The ideas that guide this topic rest on the search for unimodular multilinear forms, that is, multilinear forms with coefficients ±1, with the lowest possible norm. By using probabilistic methods, a family of inequalities that yield unimodular multilinear forms with “small norm” emerges: these are the so-called Kahane–Salem–Zygmund inequalities (KSZ inequalities for short). For the case of bilinear forms, an inequality of KSZ type was independently obtained by Bennett in 1977 in a more general setting that allows different dimensions for the spaces that form the domain of the bilinear form. It occurs that the non-deterministic approach, although very effective with regard to establishing the optimality of the exponents involved in these inequalities, provides very imprecise constants. By means of analytical results we prove that the constants are, in some cases, asymptotically dominated by 1. In addition, we provide more accurate estimates of universal domination for these cases previously known. The results and techniques from this investigation are applied to the Gale–Berlekamp switching game and allow us to improve some known estimates regarding the solutions of the game. In contrast, the best estimates for Gale–Berlekamp switching game are used to present a more accurate universal domination of the consntants in the Bennett inequality.