Home A quantitative stability result for the sphere packing problem in dimensions 8 and 24
Article
Licensed
Unlicensed Requires Authentication

A quantitative stability result for the sphere packing problem in dimensions 8 and 24

  • Károly J. Böröczky ORCID logo EMAIL logo , Danylo Radchenko and João P. G. Ramos
Published/Copyright: January 30, 2024

Abstract

We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is ε close to satisfying the optimal density, then it is, in a suitable sense, close to the E 8 and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large “frame” through which our packing locally looks like E 8 or Λ 24 . Our methods make explicit use of the magic functions constructed in [M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 2017, 3, 991–1015] in dimension 8 and in [H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 2017, 3, 1017–1033] in dimension 24, together with results of independent interest on the abstract stability of the lattices E 8 and Λ 24 .

Funding statement: J. P. G. Ramos acknowledges support by the ERC grant RSPDE 721675, and K. J. Böröczky ackowledges the hospitality of ETH Zurich where part of the research was done and the support by NKFIH grant 132002.

Acknowledgements

All authors would like to express their deepest gratitude towards the anonymous referee for indicating how to prove the current of version of Theorems 1.3 and 1.4 through adapting the techniques from [6].

References

[1] R. Bhatia and K. Mukherjea, Variation of the unitary part of a matrix, SIAM J. Matrix Anal. Appl. 15 (1994), no. 3, 1007–1014. 10.1137/S0895479892243237Search in Google Scholar

[2] H. F. Blichfeldt, The minimum values of positive quadratic forms in six, seven and eight variables, Math. Z. 39 (1935), no. 1, 1–15. 10.1007/BF01201341Search in Google Scholar

[3] K. Böröczky, Jr., Finite packing and covering, Cambridge Tracts in Math., Cambridge University, Cambridge 2004. 10.1017/CBO9780511546587Search in Google Scholar

[4] E. Caglioti, F. Golse and M. Iacobelli, Quantization of probability distributions and gradient flows in space dimension 2, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 6, 1531–1555. 10.1016/j.anihpc.2017.12.003Search in Google Scholar

[5] H. Cohn and N. Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. 10.4007/annals.2003.157.689Search in Google Scholar

[6] H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2) 170 (2009), no. 3, 1003–1050. 10.4007/annals.2009.170.1003Search in Google Scholar

[7] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. 10.4007/annals.2017.185.3.8Search in Google Scholar

[8] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, Universal optimality of the E 8 and Leech lattices and interpolation formulas, Ann. of Math. (2) 196 (2022), no. 3, 983–1082. 10.4007/annals.2022.196.3.3Search in Google Scholar

[9] H. Cohn and N. Triantafillou, Dual linear programming bounds for sphere packing via modular forms, Math. Comp. 91 (2021), no. 333, 491–508. 10.1090/mcom/3662Search in Google Scholar

[10] J. H. Conway, R. A. Parker and N. J. A. Sloane, The covering radius of the Leech lattice, Proc. Roy. Soc. Lond. Ser. A 380 (1982), no. 1779, 261–290. 10.1098/rspa.1982.0042Search in Google Scholar

[11] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss. 290, Springer, New York 1999. 10.1007/978-1-4757-6568-7Search in Google Scholar

[12] N. Elkies, Yet another proof of the uniqueness of the E8 lattice, Note available at https://people.math.harvard.edu/~elkies/Misc/E8.pdfhttps://people.math.harvard.edu/ elkies/Misc/E8.pdf. Search in Google Scholar

[13] L. Fejes, Über einen geometrischen Satz, Math. Z. 46 (1940), 83–85. 10.1007/BF01181430Search in Google Scholar

[14] G. Fejes Tóth, A stability criterion to the moment theorem, Studia Sci. Math. Hungar. 38 (2001), 209–224. 10.1556/sscmath.38.2001.1-4.14Search in Google Scholar

[15] C. F. Gauss, Untersuchungen über die Eigenschaften der positiven ternaren quadratischen Formen von Ludwig August Seeber, J. reine angew. Math. 20 (1840), 312–320. 10.1515/crll.1840.20.312Search in Google Scholar

[16] R. L. Griess, Jr., Positive definite lattices of rank at most 8, J. Number Theory 103 (2003), no. 1, 77–84. 10.1016/S0022-314X(03)00107-0Search in Google Scholar

[17] H. Groemer, Existenzsätze für Lagerungen im Euklidischen Raum, Math. Z. 81 (1963), 260–278. 10.1007/BF01111546Search in Google Scholar

[18] T. C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), no. 3, 1065–1185. 10.4007/annals.2005.162.1065Search in Google Scholar

[19] T. Hales, M. Adams, G. Bauer, T. D. Dang, J. Harrison, H. Le Truong, C. Kaliszyk, V. Magron, S. McLaughlin, T. T. Nguyen, Q. T. Nguyen, T. Nipkow, S. Obua, J. Pleso, J. Rute, A. Solovyev, T. H. A. Ta, N. T. Tran, T. D. Trieu, J. Urban, K. Vu and R. Zumkeller, A formal proof of the Kepler conjecture, Forum Math. Pi 5 (2017), Article ID e2. 10.1017/fmp.2017.1Search in Google Scholar

[20] G. A. Kabatjanskiĭ and V. I. Levenšteĭn, Bounds for packings on the sphere and in space (in Russian), Problemy Peredači Informacii 14 (1978), no. 1, 3–25; translation in Probl. Inf. Transm. 14 (1978), 1–17. Search in Google Scholar

[21] J. Kepler, On the six-cornered snowflake, Clarendon Press, Oxford 1966. Search in Google Scholar

[22] A. Korkine and G. Zolotareff, Sur les formes quadratiques positives quaternaires, Math. Ann. 5 (1872), no. 4, 581–583. 10.1007/BF01442912Search in Google Scholar

[23] A. Korkinge and G. Zolotareff, Sur les formes quadratiques positives, Math. Ann. 11 (1877), no. 2, 242–292. 10.1007/BF01442667Search in Google Scholar

[24] J. L. Lagrange, Solutions analytiques de quelques problemes sur les pyramides triangulaires, Académie royale des sciences et belles lettres, Brussel 1973. Search in Google Scholar

[25] L. J. Mordell, The definite quadratic forms in eight variables with determinant unity, J. Math. Pures Appl. (9) 17 (1938), 41–46. Search in Google Scholar

[26] H.-V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskriminante 1, J. Number Theory 5 (1973), 142–178. 10.1016/0022-314X(73)90068-1Search in Google Scholar

[27] B. Segre and K. Mahler, On the densest packing of circles, Amer. Math. Monthly 51 (1944), 261–270. 10.1080/00029890.1944.11999084Search in Google Scholar

[28] A. Thue, On some geometric number-theoretic theorems (in Danish), Forhandlingerne Skand. Naturforskeres 14 (1892), 352–353. Search in Google Scholar

[29] A. Thue, Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene, Norske Videnskabs-Selskabets Skrifter 1 (1910), 1–9. Search in Google Scholar

[30] A. Venkatesh, A note on sphere packings in high dimension, Int. Math. Res. Not. IMRN 2013 (2013), no. 7, 1628–1642. 10.1093/imrn/rns096Search in Google Scholar

[31] M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), no. 3, 991–1015. 10.4007/annals.2017.185.3.7Search in Google Scholar

Received: 2023-02-27
Revised: 2023-07-14
Published Online: 2024-01-30
Published in Print: 2024-03-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 21.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2024-0002/html?lang=en
Scroll to top button