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

The Kahane–Salem–Zygmund inequality for multilinear forms in ${\mathrm{\ell }}_{\mathrm{\infty }}$ spaces claims that, for all positive integers $m,n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_m$, 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)},\mathrm{.}\mathrm{.}\mathrm{.},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{,}$$ satisfying $$\mathrm{\parallel }A\mathrm{\parallel }\le C_m\max \{n_1^{1\mathrm{/}2},\dots ,n_m^{1\mathrm{/}2}\}{\prod }_{j=1}^m{n}_j^{1\mathrm{/}2}\text{,}$$ for $$C_m\le \kappa \sqrt{m\log m}\sqrt{m!}$$ and a certain $\kappa >0.$ Our main result shows that given any $ϵ>0$ and any positive integer $m,$ there exists a positive integer $N$ such that $$C_m<1+ϵ\text{,}$$ when we consider $n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_m>N$. 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.