Equidistribution of primitive lattices in ℝn

DOI10.1093/QMATH/HAAD008arXiv2012.04508OpenAlexW4380090676MaRDI QIDQ6183304

Publication date: 26 January 2024

Published in: The Quarterly Journal of Mathematics (Search for Journal in Brave)

Abstract: We count primitive lattices of rank

d

inside

m a t h b b Z^{n}

as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its projection to the Grassmannian; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude joint equidistribution of these parameters. In addition to the primitive

d

-lattices themselves, we also consider their orthogonal complements in

m a t h b b Z^{n}

, and show that the equidistribution occurs jointly for primitive lattices and their orthogonal complements. Finally, our asymptotic formulas for the number of primitive lattices include an explicit error term.

Full work available at URL: https://arxiv.org/abs/2012.04508

Mathematics Subject Classification ID

Lattices and convex bodies in (n) dimensions (aspects of discrete geometry) (52C07) Asymptotic results on counting functions for algebraic and topological structures (11N45) Lattices and convex bodies (number-theoretic aspects) (11H06) Well-distributed sequences and other variations (11K36)