论文标题
Kahane-Salem的常数 - Zygmund不等式渐近地限制了$ 1 $
Constants of the Kahane--Salem--Zygmund inequality asymptotically bounded by $1$
论文作者
论文摘要
Kahane--salem-- $ \ ell _ {\ infty} $ spaces中多线性形式的不平等不平等，声称，对于所有积极的整数$ m，n_ {1}，...，n_ {m}美元类型\ [a（z^{（1）}，...，z^{（m）}）= \ sum_ {j__ {1} = 1}^n_ {n_ {1}} \ cdots \ sum___ {m} {m} {m} = 1} n_ z_ {j_ {1}}^{\ left（1 \右）} \ cdots z_ { n_ {1}^{1/2}，\ ldots，n_ {m}^{1/2} \ right \} {\ textStyle \ prod \ limits_ { c_ {m} \leqκ\ sqrt {m \ log m} \ sqrt {m！} \]和一定的$κ>0。$我们的主要结果表明，给定任何$ε> 0 $和任何积极的integer $ m，$ $ $，$，$，$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n WE，Wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer wer we c_} c {m} $ n_ {1}，...，n_ {m}> n $。此外，虽然Kahane-Salem- Zygmund的原始证明依赖于高度非确定性的论点，但我们的方法是建设性的。我们还为G. Bennett在1977年证明的相关非确定性不平等的常数提供了相同的渐近约束（在某些情况下显示为最佳）。给出了Berlekamp转换游戏的应用。
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 \[ C_{m}\leqκ\sqrt{m\log m}\sqrt{m!} \] 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 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.
