The Equidistribution of Grids of Rings of Integers in Number Fields of Degrees 3,4 and 5

It was shown by M. Bhargava and P. Harron that for $n=3,4,5$, the shapes of rings of integers of $S_n$-number fields of degree $n$ become equidistributed in the space of shapes when the fields are ordered by discriminant. Instead of shapes, we correspond grids to each number field, which preserve more of the number fields’ data. The space of grids is a fiber bundle over the space of shapes. We strengthen Bhargava-Harron’s result by proving that the grids of rings of integers of $S_n$-number fields become equidistributed in the space of grids.