Most totally real fields do not have universal forms or the Northcott property - Nicolas Daans; Vítězslav Kala; Siu Hang Man; Martin Widmer; Pavlo Yatsyna (Proceedings of the National Academy of Sciences, 2025)
We show that, in the space of all totally real fields equipped with the constructible topology, the set of fields that admit a universal quadratic form, or have the Northcott property, is meager. The main tool is a theorem on the number of square classes of totally positive units represented by a quadratic lattice of a given rank.