PreprintCC BY 4.0arXiv
On two counterexamples in the geometry of numbers
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. The counterexample also disproves the natural product formula for unrestricted congruent packings. 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.
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BibTeX
@misc{gargava2026counterexamples,
title = {{On two counterexamples in the geometry of numbers}},
author = {Nihar Gargava},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.11695},
eprint = {2607.11695},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11695},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On two counterexamples in the geometry of numbers
AU - Nihar Gargava
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.11695
UR - https://arxiv.org/abs/2607.11695
AB - 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. The counterexample also disproves the natural product formula for unrestricted congruent packings. 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.
ER -