Constants of the Kahane–Salem–Zygmund inequality asymptotically bounded by 1

The Kahane–Salem–Zygmund inequality for multilinear forms in ?? spaces claims that, all positive integers m,n1,...,nm, there exists an m-linear form A:??n1×?×??nm?K (K=R or C) of the typeA(z(1),...,z(m))=?j1=1n1??jm=1nm±zj1(1)?zjm(m), satisfying?A??Cmmax?{n11/2,…,nm1/2}?j=1mnj1/2, forCm??mlog?mm! and a certain ?>0. Our main result shows that given any ?>0 integer m, N such thatCm<1+?, when we consider n1,...,nm>N. In addition, while original proof relies on highly non-deterministic arguments, our approach is constructive. We also provide same asymptotic bound (which shown to be optimal some cases) constant related proved by G. Bennett 1977. Applications Berlekamp's switching game are given.