We consider the problem of finding dense rank one module sublattices in module lattices of a given rank, where by module lattice we mean a lattice obtained as R-linear combinations of vectors with coefficients in a number field K, with ring of integers 𝒪 K , and R ⊆ 𝒪 K an order.We provide generic upper bounds on the normalized volume of rank one module sublattices for K totally real or CM, specific bounds for quadratic extensions and some cyclotomic extensions, as well as exact values for some norm Euclidean quadratic extensions.
Frederique Oggier (Fri,) studied this question.