Asymptotics of $n$-universal lattices over number fields

TL;DR

This paper derives an explicit asymptotic formula for the minimal ranks of n-universal lattices over totally real number fields and proves finiteness results for fields admitting such lattices or criteria sets of bounded size.

Contribution

It provides the first explicit asymptotic formula for minimal ranks and establishes finiteness and computability results for totally real fields with bounded universal lattice properties.

Findings

Explicit asymptotic formula for minimal ranks of n-universal lattices.

Finiteness of totally real fields with bounded universal lattice rank or criteria set size.

All such fields are effectively computable.

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.