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arXiv · 2510.26652

Asymptotics of $n$-universal lattices over number fields

Dayoon Park·Robin Visser·Pavlo Yatsyna·Jongheun Yoon
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Abstract

We prove an explicit asymptotic formula for the logarithm of the minimal ranks of $n$-universal lattices over the ring of integers of totally real number fields. We also show that, for any constant $C > 0$ and $n \geq 3$, there are only finitely many totally real fields with an $n$-universal lattice of rank at most $C$, with all such fields being effectively computable. Similarly, for any $n \geq 3$, we show that there are only finitely many totally real fields admitting an $n$-universal criterion set of size at most $C$, with all such fields likewise being effectively computable.

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Dayoon Park, Robin Visser, Pavlo Yatsyna, Jongheun Yoon. 2025-10-30. Asymptotics of $n$-universal lattices over number fields. https://arxiv.org/abs/2510.26652

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