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

The Kahane–Salem–Zygmund inequality for multilinear forms in ℓ∞ spaces claims that, for all positive integers m,n1,...,nm, there exists an m-linear form A:ℓ∞n1×⋯×ℓ∞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≤κmlog⁡mm! and a certain κ>0. Our main result shows that given any ϵ>0 and any positive integer m, there exists a positive integer N such thatCmN. In addition, while the original proof of the Kahane–Salem–Zygmund relies on highly non-deterministic arguments, our approach is constructive. We also provide the same asymptotic bound (which is shown to be optimal in some cases) for the constant of a related non-deterministic inequality proved by G. Bennett in 1977. Applications to Berlekamp's switching game are given.