M. Bhargava and P. Harron demonstrated that for
, the shapes of rings
of integers of
-number
fields are equidistributed in the space of shapes when ordered by discriminant. In this
paper, we construct grids as a refinement of shapes, capturing additional geometric
data about the rings of integers. Grids form a fiber bundle over the space of shapes,
offering a richer perspective on number fields. We extend Bhargava and Harron’s
results by proving that grids are also equidistributed in their respective space
according to the Haar measure, providing a deeper understanding of the
distributional properties of number fields.
Keywords
equidistribution, number fields, rings of integers, shapes,
grids, lattices, homogeneous spaces, discriminant ordering
