关于数的几何中的两个反例
On two counterexamples in the geometry of numbers
浏览论文内容
中文总结 AI 辅助
该研究给出八维和九维中两个优化问题反例，一是关于卡塞尔提出的笛卡尔积问题，二是萨纳克提出的问题，通过精确及数值反例分别反驳相关结论，还反驳了无限制全等填充的自然积公式。
中文摘要 AI 辅助
我们给出了八维和九维中两个优化问题的反例。一是卡塞尔提出的关于临界行列式的笛卡尔积问题，后由宗针对格填充以及允许平移但不允许旋转的填充重新表述：相应的积不等式是否总是等式。二是萨纳克提出并由邱表述为猜想的问题：在单位体积的平坦环面中，高度是否由使其最短非零向量长度最大化的格最小化。第一个反例精确，还反驳了无限制全等填充的自然积公式。第二个是数值性反例，但在合理的浮点精度范围内。
英文摘要
We give counterexamples to two optimization problems in dimensions eight and nine. 1. The Cartesian-product problem posed by Cassels for critical determinants and later formulated by Zong for lattice packings and for packings allowing translations but not rotations: whether the corresponding product inequalities are always equalities. 2. A question raised by Sarnak and formulated as a conjecture in Chiu: whether, among unit-volume flat tori, height is minimized by a lattice maximizing the length of its shortest nonzero vector. The first counterexample is exact and also disproves the natural product formula for unrestricted congruent packings. The second is numerical but within reasonable floating-point accuracy.
