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

TL;DR

This paper extends previous results on the distribution of number fields by proving that the more detailed structure of grids of rings of integers in $S_n$-number fields of degrees 3, 4, and 5 also becomes equidistributed, providing a finer understanding of their distribution.

Contribution

The paper proves the equidistribution of grids of rings of integers in $S_n$-number fields, refining prior results on shapes by considering more detailed grid structures.

Findings

Grids of rings of integers become equidistributed in the space of grids.

Strengthens previous shape distribution results by considering more detailed data.

Provides a deeper understanding of the distribution of number fields.

Abstract

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.