Equidistribution of primitive lattices in ℝn

Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 74(4), 1253–1294.

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Author
Horesh, TalISTA; Karasik, Yakov
Department
Abstract
We count primitive lattices of rank d inside Zn as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subspace 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 the joint equidistribution of these parameters. In addition to the primitive d-lattices Λ themselves, we also consider their orthogonal complements in Zn⁠, A1⁠, and show that the equidistribution occurs jointly for Λ and A1⁠. Finally, our asymptotic formulas for the number of primitive lattices include an explicit bound on the error term.
Publishing Year
Date Published
2023-12-01
Journal Title
Quarterly Journal of Mathematics
Acknowledgement
This work was done when both authors were visiting Institute of Science and Technology (IST) Austria. T.H. was being supported by Engineering and Physical Sciences Research Council grant EP/P026710/1. Y.K. had a great time there and is grateful for the hospitality. The appendix to this paper is largely based on a mini course T.H. had given at IST in February 2020.
Volume
74
Issue
4
Page
1253-1294
ISSN
eISSN
IST-REx-ID

Cite this

Horesh T, Karasik Y. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 2023;74(4):1253-1294. doi:10.1093/qmath/haad008
Horesh, T., & Karasik, Y. (2023). Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. Oxford University Press. https://doi.org/10.1093/qmath/haad008
Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in ℝn.” Quarterly Journal of Mathematics. Oxford University Press, 2023. https://doi.org/10.1093/qmath/haad008.
T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,” Quarterly Journal of Mathematics, vol. 74, no. 4. Oxford University Press, pp. 1253–1294, 2023.
Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly Journal of Mathematics. 74(4), 1253–1294.
Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in ℝn.” Quarterly Journal of Mathematics, vol. 74, no. 4, Oxford University Press, 2023, pp. 1253–94, doi:10.1093/qmath/haad008.
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2024-01-02
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