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

Abstract: The Kahane--Salem--Zygmund inequality for multilinear forms in $\ell_{\infty}$ spaces claims that, for all positive integers $m,n_{1},...,n_{m}$, there exists an $m$-linear form $A\colon\ell_{\infty}^{n_{1}}\times\cdots\times \ell_{\infty}^{n_{m}}\longrightarrow\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) of the type [ A(z^{(1)},...,z^{(m)})=\sum_{j_{1}=1}^{n_{1}}\cdots\sum_{j_{m}=1}^{n_{m}}\pm z_{j_{1}}^{\left( 1\right) }\cdots z_{j_{m}}^{\left( m\right) }\text{,} ] satisfying [ \Vert A\Vert\leq C_{m}\max\left{ n_{1}^{1/2},\ldots,n_{m}^{1/2}\right} {\textstyle\prod\limits_{j=1}^{m}}n_{j}^{1/2}\text{,} ] for [ C_{m}\leq\kappa\sqrt{m\log m}\sqrt{m!} ] and a certain $\kappa>0.$ Our main result shows that given any $\epsilon>0$ and any positive integer $m,$ there exists a positive integer $N$ such that [ C_{m}<1+\epsilon\text{,} ] when we consider $n_{1},...,n_{m}>N$. In addition, while the original proof of the Kahane--Salem--Zygmund relies in 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.