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 $A:{\ell }_{\infty }^{n_1}\times \cdots \times {\ell }_{\infty }^{n_m}⟶\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ 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 \[ C_{m}\leqκ\sqrt{m\log m}\sqrt{m!} \] 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 \[ C_{m}<1+ε\text{,} \] 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.

This version incorporates the results presented in arXiv:2104.10765 (the preprint arXiv:2104.10765 was withdrawn)

References in corpus (1)

• Subgaussian Kahane-Salem-Zygmund inequalities in Banach spaces