AI 中文总结
本文通过构造高维布尔对族，证明柯尼希常数为1，证伪了相关高维猜想，对格罗滕迪克常数的相关研究给出了否定回答。
AI 中文摘要
对于每个整数N≥1，考虑归一化柯尼希双线性形式$B_{\mathrm K}:L_\infty(\mathbb R^N)\times L_\infty(\mathbb R^N)\to\mathbb R$，其定义为：$B_{\mathrm K}(f,g):=\frac{1}{(\sqrt{2}\pi)^N} \iint_{\mathbb R^N\times\mathbb R^N} f(x)g(y)e^{-(\lVert x\rVert^2+\lVert y\rVert^2)/2} \sin\langle x,y\rangle\\,\mathrm d x\\,\mathrm d y$。我们将柯尼希常数定义为：$\mathfrak K_{\mathrm K}:=\sup_{N\geq1}\sup_{\substack{f,g:\mathbb R^N\to\{\pm1\}\\ f,g\\ \mathrm{measurable}}}B_{\mathrm K}(f,g)$。对该双线性形式的研究源于确定格罗滕迪克常数精确值的努力，König猜想其精确值应为一维半空间对应的$B_{\mathrm K}(\operatorname{sgn}(x_1),\operatorname{sgn}(x_1))=\frac{2}{\pi}\log(1+\sqrt{2})$。若该猜想成立，结合Krivine的经典上界，将确定格罗滕迪克常数的精确值。Braverman、Makarychev、Makarychev和Naor在突破性研究中于二维维度证伪了König的猜想，并利用其反例首次实现了对Krivine界的严格改进。该研究中提出的一个问题试图通过高维下源于柯尼希双线性形式的交替Krivine舍入方案确定格罗滕迪克常数。近期，Li等人构造了高维实例，表明$\mathfrak K_{\mathrm K}\ge 0.59357$。一个初等傅里叶论证给出$\mathfrak K_{\mathrm K}\le 1$，且排除了有限维度下取等的可能。本文通过构造高维下的布尔对族，证明$\mathfrak K_{\mathrm K}=1$，这对Braverman等人提出的问题的高维方面给出了否定回答。
英文摘要
For each $N\geq1$, consider the normalized König bilinear form $B_{\mathrm K}:L_\infty(\mathbb R^N)\times L_\infty(\mathbb R^N)\to\mathbb R$ given by \[ B_{\mathrm K}(f,g):=\frac{1}{(\sqrt{2}π)^N} \iint_{\mathbb R^N\times\mathbb R^N} f(x)g(y)e^{-(\lVert x\rVert^2+\lVert y\rVert^2)/2} \sin\langle x,y\rangle\,\mathrm d x\,\mathrm d y, \] We define the König constant by \[ \mathfrak K_{\mathrm K}:=\sup_{N\geq1}\sup_{\substack{f,g:\mathbb R^N\to\{\pm1\}\\ f,g\ \mathrm{measurable}}}B_{\mathrm K}(f,g). \] The study of this bilinear form arose from efforts to determine the exact value of the Grothendieck constant. König~\cite{KONIG} conjectured that the sharp value should instead be given by the one-dimensional half-spaces $B_{\mathrm K}(\operatorname{sgn}(x_1),\operatorname{sgn}(x_1))=\frac{2}π\log(1+\sqrt{2})$. A positive answer to this conjecture, together with a classical upper bound of Krivine \cite{KRIVINE}, would determine the exact value of the Grothendieck constant. In a breakthrough~\cite{BMMN}, Braverman, Makarychev, Makarychev, and Naor disproved König's conjecture already in dimension two and used their counterexamples to obtain the first strict improvement over Krivine's bound. One question in \cite{BMMN} attempts to determine the Grothendieck constant through alternating Krivine rounding schemes arising from König's bilinear form in high dimension. More recently, Li et al.~\cite{LISK} constructed high-dimensional examples showing that $\mathfrak K_{\mathrm K}\ge 0.59357$. An elementary Fourier argument gives $\mathfrak K_{\mathrm K}\le 1$ and excludes equality for every finite-dimension. In this paper, we prove that $\mathfrak K_{\mathrm K}=1$ by constructing a family of Boolean pairs in high dimensions. In particular, this gives a negative answer to the high-dimensional aspect of the question in \cite{BMMN}.
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