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The equidistribution of grids of rings of integers in number fields of degrees $3$, $4$ and $5$

Yuval Yifrach
Vol. 15 (2026), No. 1, 43–72
DOI: 10.2140/cnt.2026.15.43
Abstract

M. Bhargava and P. Harron demonstrated that for n = 3,4,5, the shapes of rings of integers of Sn-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.

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Keywords
equidistribution, number fields, rings of integers, shapes, grids, lattices, homogeneous spaces, discriminant ordering
Mathematical Subject Classification
Primary: 06B99, 60B15
Milestones
Received: 16 April 2024
Revised: 4 March 2026
Accepted: 19 March 2026
Published: 17 April 2026
Authors
Yuval Yifrach
University of Zurich
Zurich
Switzerland
© 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).