发表机构
Carnegie Mellon University（卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造反例否证了关于两个超平面的Ramos猜想，并确定了保证多个测度存在公共二超平面等分的最小维数，同时揭示了与相位恢复中正定测量数目的联系。
AI 中文摘要
对于每个$n\ge2$，我们在$\mathbb{R}^{6n-3}$上构造$4n-2$个非退化高斯测度，它们不能被两个仿射超平面同时等分。这否证了关于两个超平面的Ramos猜想。结合已知的上界，该构造表明，对于每个$s\ge3$，$3\cdot2^{s-1}-2$是保证$2^s-2$个绝对连续概率测度具有公共二超平面等分的最小维数。我们用相位恢复所需的正定二次测量的最小数目来刻画高斯等分阈值。修正的复多项式乘法在每一个偶数维$r\ge4$中给出$2r-2$个正定测量。当$r=2^k+2$（$k\ge1$）时，这个数目是最优的。
英文摘要
For every $n\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\ge3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\ge4$. This number is optimal when $r=2^k+2$, $k\ge1$.
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