  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.
