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Moments of the number of points in a bounded set for number field lattices (2308.15275v2)

Published 29 Aug 2023 in math.NT, cs.IT, math.DS, and math.IT

Abstract: We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $\omega_K$ the number of roots of unity in $K$, we show that for lattices of large enough dimension the moments of the number of $\omega_K$-tuples of lattice points converge to those of a Poisson distribution of mean $V/\omega_K$. This extends work of Rogers for $\mathbb{Z}$-lattices. What is more, we show that this convergence can also be achieved by increasing the degree of the number field $K$ as long as $K$ varies within a set of number fields with uniform lower bounds on the absolute Weil height of non-torsion elements.

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References (38)
  1. Carl L. Siegel. A mean value theorem in geometry of numbers. Ann. of Math. (2), 46(2):340–347, 1945.
  2. Edmund Hlawka. Zur Geometrie der Zahlen. Math. Z., 49(1):285–312, 1943.
  3. Stephanie Vance. Improved sphere packing lower bounds from Hurwitz lattices. Adv. Math., 227(5):2144–2156, 2011.
  4. Akshay Venkatesh. A note on sphere packings in high dimension. Int. Math. Res. Not. IMRN, 2013(7):1628–1642, 2013.
  5. Nihar P. Gargava. Lattice packings through division algebras. Math. Z., 303(1):1–32, 2023.
  6. Claude A. Rogers. The moments of the number of points of a lattice in a bounded set. Philos. Trans. Roy. Soc. A, 248(945):225–251, 1955.
  7. Claude A. Rogers. Mean values over the space of lattices. Acta Math., 94:249–287, 1955.
  8. Claude A. Rogers. The number of lattice points in a set. Proc. Lond. Math. Soc. (3), 6(2):305–320, 1956.
  9. Seungki Kim. Adelic Rogers integral formula. arXiv:2205.03138, 2022.
  10. André Weil. Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math., 113:1–87, 1965.
  11. Nathan Hughes. Mean values over lattices in number fields and effective Diophantine approximation. arXiv:2306.02499, 2023.
  12. Claude A. Rogers. Lattice coverings of space: the Minkowski–Hlawka theorem. Proc. Lond. Math. Soc. (3), 8(3):447–465, 1958.
  13. New bounds on the density of lattice coverings. J. Amer. Math. Soc., 35(1):295–308, 2022.
  14. Logarithm laws for flows on homogeneous spaces. Invent. Math., 138:451–494, 1998.
  15. Logarithm laws for unipotent flows, I. J. Mod. Dyn., 3(3):359–378, 2009.
  16. Poisson approximation and Weibull asymptotics in the geometry of numbers. Trans. Amer. Math. Soc., 376(3):2155–2180, 2023.
  17. Two central limit theorems in Diophantine approximation. arXiv:2306.02304, 2023.
  18. Pseudorandomness of ring-LWE for any ring and modulus. In STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 461–473. ACM, New York, 2017.
  19. Mildly short vectors in cyclotomic ideal lattices in quantum polynomial time. J. ACM, 68(2):1–26, 2021.
  20. A fully classical LLL algorithm for modules. Cryptology ePrint Archive, Paper 2022/1356, 2022. https://eprint.iacr.org/2022/1356.
  21. Wolfgang M. Schmidt. On heights of algebraic subspaces and Diophantine approximations. Ann. of Math. (2), 85(3):430–472, 1967.
  22. Application of automorphic forms to lattice problems. J. Math. Crypt., 16(1):156–197, 2022.
  23. Andrzej Schinzel. On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith., 24(4):385–399, 1973.
  24. A lower bound for the height in abelian extensions. J. Number Theory, 80(2):260–272, 2000.
  25. Around the Unit Circle: Mahler Measure, Integer Matrices and Roots of Unity. Universitext. Springer International Publishing, 2021.
  26. On fields with Property (B). Proc. Amer. Math. Soc., 142(6):1893–1910, 2014.
  27. Michel Langevin. Minorations de la maison et de la mesure de Mahler de certains entiers algébriques. C. R. Acad. Sci. Paris Sér. I Math., 303(12):523–526, 1986.
  28. Lukas Pottmeyer. Small totally p𝑝pitalic_p-adic algebraic numbers. Int. J. Number Theory, 14(10):2687–2697, 2018.
  29. A note on heights in certain infinite extensions of ℚℚ\mathbb{Q}blackboard_Q. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 12(1):5–14, 2001.
  30. Philipp Habegger. Small height and infinite nonabelian extensions. Duke Math. J., 162(11):2027–2076, 2013.
  31. Edward Dobrowolski. On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith., 34(4):391–401, 1979.
  32. Paul Voutier. An effective lower bound for the height of algebraic numbers. Acta Arith., 74(1):81–95, 1996.
  33. Thomas Ange. Le théorème de Schanuel dans les fibrés adéliques Hermitiens. Manuscripta Math., 144(3-4):565–608, 2014.
  34. David W. Boyd. Variations on a theme of Kronecker. Canad. Math. Bull., 21(2):129–133, 1978.
  35. Claude A. Rogers. Two integral inequalities. J. Lond. Math. Soc., 31(2):235–238, 1956.
  36. André Weil. Discontinuous subgroups of classical groups: lectures. University of Chicago, 1958.
  37. Armand Borel. Introduction to arithmetic groups, volume 73. American Mathematical Soc., 2019.
  38. Armand Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann. of Math, 75(3):485–535, 1962.
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