Two Counterexamples in the Geometry of Numbers
Statement
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.
Context
Refutes two separate stated problems at once, one of them a question of Sarnak.
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The acknowledgements state the counterexamples were found using ChatGPT, and that the final paper is partly the author's own writing and partly written with the help of Codex.
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