On two counterexamples in the geometry of numbers
arXiv:2607.11695
The paper presents two counterexamples in dimensions eight and nine: one disproving a product‑equality conjecture for critical determinants and lattice packings, and another showing that the height of unit‑volume flat tori is not always minimized by the lattice with the longest shortest vector.
Abstract
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.
Ancillary files contains reproduction of computations