Dual linear programming bounds for sphere packing via modular forms
HTML articles powered by AMS MathViewer
- by Henry Cohn and Nicholas Triantafillou;
- Math. Comp. 91 (2022), 491-508
- DOI: https://doi.org/10.1090/mcom/3662
- Published electronically: July 30, 2021
- HTML | PDF | Request permission
Abstract:
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions $8$ and $24$, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions $12$, $16$, $20$, $28$, and $32$. More generally, we provide a systematic technique for proving separations of this sort.References
- N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat, and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, J. High Energy Phys. 2020, no. 12, 066, 44 pp. arXiv:2006.02560, DOI:10.1007/JHEP12(2020)066.
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Jean Bourgain, Laurent Clozel, and Jean-Pierre Kahane, Principe d’Heisenberg et fonctions positives, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 4, 1215–1232 (French, with English and French summaries). MR 2722239, DOI 10.5802/aif.2552
- Henri Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, vol. 240, Springer, New York, 2007. MR 2312338
- Henry Cohn, New upper bounds on sphere packings. II, Geom. Topol. 6 (2002), 329–353. MR 1914571, DOI 10.2140/gt.2002.6.329
- Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. MR 1973059, DOI 10.4007/annals.2003.157.689
- Henry Cohn and Felipe Gonçalves, An optimal uncertainty principle in twelve dimensions via modular forms, Invent. Math. 217 (2019), no. 3, 799–831. MR 3989254, DOI 10.1007/s00222-019-00875-4
- Henry Cohn and Abhinav Kumar, Universally optimal distribution of points on spheres, J. Amer. Math. Soc. 20 (2007), no. 1, 99–148. MR 2257398, DOI 10.1090/S0894-0347-06-00546-7
- Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. MR 3664817, DOI 10.4007/annals.2017.185.3.8
- H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, Universal optimality of the $E_8$ and Leech lattices and interpolation formulas, preprint, arXiv:1902.05438, 2019.
- H. Cohn and N. Triantafillou, Data for “Dual linear programming bounds for sphere packing via modular forms”, data set, DSpace@MIT, 2021. https://hdl.handle.net/1721.1/130355.
- Henry Cohn and Yufei Zhao, Sphere packing bounds via spherical codes, Duke Math. J. 163 (2014), no. 10, 1965–2002. MR 3229046, DOI 10.1215/00127094-2738857
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
- Fred Diamond and John Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133. MR 1357209
- Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
- Thomas C. Hales, A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), no. 3, 1065–1185. MR 2179728, DOI 10.4007/annals.2005.162.1065
- Thomas Hales, Mark Adams, Gertrud Bauer, Tat Dat Dang, John Harrison, Le Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat Thang Nguyen, Quang Truong Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, Thi Hoai An Ta, Nam Trung Tran, Thi Diep Trieu, Josef Urban, Ky Vu, and Roland Zumkeller, A formal proof of the Kepler conjecture, Forum Math. Pi 5 (2017), e2, 29. MR 3659768, DOI 10.1017/fmp.2017.1
- E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann. 112 (1936), no. 1, 664–699 (German). MR 1513069, DOI 10.1007/BF01565437
- David de Laat, Fernando Mário de Oliveira Filho, and Frank Vallentin, Upper bounds for packings of spheres of several radii, Forum Math. Sigma 2 (2014), Paper No. e23, 42. MR 3264261, DOI 10.1017/fms.2014.24
- Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
- Stephen D. Miller and Wilfried Schmid, Summation formulas, from Poisson and Voronoi to the present, Noncommutative harmonic analysis, Progr. Math., vol. 220, Birkhäuser Boston, Boston, MA, 2004, pp. 419–440. MR 2036579
- Hans D. Mittelmann and Frank Vallentin, High-accuracy semidefinite programming bounds for kissing numbers, Experiment. Math. 19 (2010), no. 2, 175–179. MR 2676746, DOI 10.1080/10586458.2010.10129070
- Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 256993
- Masami Ohta, On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms, J. Reine Angew. Math. 463 (1995), 49–98. MR 1332907, DOI 10.1515/crll.1995.463.49
- Danylo Radchenko and Maryna Viazovska, Fourier interpolation on the real line, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 51–81. MR 3949027, DOI 10.1007/s10240-018-0101-z
- Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.8), 2019, http://www.sagemath.org.
- A. Scardicchio, F. H. Stillinger, and S. Torquato, Estimates of the optimal density of sphere packings in high dimensions, J. Math. Phys. 49 (2008), no. 4, 043301, 15. MR 2412293, DOI 10.1063/1.2897027
- William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048, DOI 10.1090/gsm/079
- A. Thue, Om nogle geometrisk-taltheoretiske Theoremer, Forhandlingerne ved de Skandinaviske Naturforskeres 14 (1892), 352–353.
- S. Torquato and F. H. Stillinger, New conjectural lower bounds on the optimal density of sphere packings, Experiment. Math. 15 (2006), no. 3, 307–331. MR 2264469, DOI 10.1080/10586458.2006.10128964
- Maryna S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), no. 3, 991–1015. MR 3664816, DOI 10.4007/annals.2017.185.3.7
Bibliographic Information
- Henry Cohn
- Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachusetts 02142
- MR Author ID: 606578
- ORCID: 0000-0001-9261-4656
- Email: cohn@microsoft.com
- Nicholas Triantafillou
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Address at time of publication: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 978350
- ORCID: 0000-0003-1464-4619
- Email: nicholas.triantafillou@gmail.com
- Received by editor(s): September 23, 2019
- Received by editor(s) in revised form: April 4, 2021
- Published electronically: July 30, 2021
- Additional Notes: The second author was partially supported by an internship at Microsoft Research New England, a National Science Foundation Graduate Research Fellowship under grant #1122374, and Simons Foundation grant #550033.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 491-508
- MSC (2020): Primary 52C17; Secondary 11H31, 11F11
- DOI: https://doi.org/10.1090/mcom/3662
- MathSciNet review: 4350546