Two Counterexamples in the Geometry of Numbers
Prior state unknown→disproved
The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated as a conjecture by Chiu, on whether height among unit-volume flat tori is minimized by a lattice maximizing its shortest nonzero vector.
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