From sphere packing to Fourier interpolation

Cohn, Henry

doi:10.1090/bull/1813

From sphere packing to Fourier interpolation
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by Henry Cohn

Bull. Amer. Math. Soc. 61 (2024), 3-22

DOI: https://doi.org/10.1090/bull/1813

Published electronically: October 6, 2023

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Abstract:

Viazovska’s solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska.

References

Anshul Adve, Density criteria for Fourier uniqueness phenomena in $\mathbb {R}^d$, arXiv:2306.07475 (2023).
Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, David de Laat, and Amirhossein Tajdini, High-dimensional sphere packing and the modular bootstrap, J. High Energy Phys. 12 (2020), Paper No. 066, 44. MR 4239386, DOI 10.1007/jhep12(2020)066
Andrew Bakan, Haakan Hedenmalm, Alfonso Montes-Rodríguez, Danylo Radchenko, and Maryna Viazovska, Fourier uniqueness in even dimensions, Proc. Natl. Acad. Sci. USA 118 (2021), no. 15, Paper No. 2023227118, 4. MR 4294062, DOI 10.1073/pnas.2023227118
Andrew Bakan, Haakan Hedenmalm, Alfonso Montes-Rodríguez, Danylo Radchenko, and Maryna Viazovska, Hyperbolic Fourier series, arXiv:2110.00148 (2021).
M. R. Best, Binary codes with a minimum distance of four, IEEE Trans. Inform. Theory 26 (1980), no. 6, 738–742. MR 596287, DOI 10.1109/TIT.1980.1056269
Andriy Bondarenko, Danylo Radchenko, and Kristian Seip, Fourier interpolation with zeros of zeta and $L$-functions, Constr. Approx. 57 (2023), no. 2, 405–461. MR 4577389, DOI 10.1007/s00365-022-09599-w
S. V. Borodachov, D. P. Hardin, and E. B. Saff, Asymptotics of best-packing on rectifiable sets, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2369–2380. MR 2302558, DOI 10.1090/S0002-9939-07-08975-7
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
Henry Cohn, A conceptual breakthrough in sphere packing, Notices Amer. Math. Soc. 64 (2017), no. 2, 102–115. MR 3587715, DOI 10.1090/noti1474
Henry Cohn, The work of Maryna Viazovska, Fields medal laudatio, 2022. arXiv:2207.06913., DOI 10.4171/ICM2022/213
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, 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
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, Universal optimality of the $E_8$ and Leech lattices and interpolation formulas, Ann. of Math. (2) 196 (2022), no. 3, 983–1082. MR 4502595, DOI 10.4007/annals.2022.196.3.3
Henry Cohn, David de Laat, and Andrew Salmon, Three-point bounds for sphere packing, arXiv:2206.15373 (2022).
J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discrete Comput. Geom. 13 (1995), no. 3-4, 383–403. MR 1318784, DOI 10.1007/BF02574051
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
Matthew de Courcy-Ireland, Maria Dostert, and Maryna Viazovska, Six-dimensional sphere packing and linear programming, arXiv:2211.09044 (2022).
P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep. 27 (1972), 272–289. MR 314545
Wolfgang Ebeling, Lattices and codes, 3rd ed., Advanced Lectures in Mathematics, Springer Spektrum, Wiesbaden, 2013. A course partially based on lectures by Friedrich Hirzebruch. MR 2977354, DOI 10.1007/978-3-658-00360-9
L. Fejes, Über einen geometrischen Satz, Math. Z. 46 (1940), 83–85 (German). MR 1587, DOI 10.1007/BF01181430
Felipe Gonçalves, Diogo Oliveira e Silva, and Stefan Steinerberger, Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, J. Math. Anal. Appl. 451 (2017), no. 2, 678–711. MR 3624763, DOI 10.1016/j.jmaa.2017.02.030
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
Thomas Hartman, Dalimil Mazáč, and Leonardo Rastelli, Sphere packing and quantum gravity, J. High Energy Phys. 12 (2019), 048, 66. MR 4075697, DOI 10.1007/jhep12(2019)048
Haakan Hedenmalm and Alfonso Montes-Rodríguez, Heisenberg uniqueness pairs and the Klein-Gordon equation, Ann. of Math. (2) 173 (2011), no. 3, 1507–1527. MR 2800719, DOI 10.4007/annals.2011.173.3.6
Haakan Hedenmalm and Alfonso Montes-Rodríguez, The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 6, 1703–1757. MR 4092897, DOI 10.4171/jems/954
Haakan Hedenmalm and Alfonso Montes-Rodríguez, The Klein-Gordon equation, the Hilbert transform and Gauss-type maps: $H^{\infty }$ approximation, J. Anal. Math. 144 (2021), no. 1, 119–190. MR 4361892, DOI 10.1007/s11854-021-0173-4
J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 45–89. MR 766960, DOI 10.1090/S0273-0979-1985-15293-0
Aleksei Kulikov, Fourier interpolation and time-frequency localization, J. Fourier Anal. Appl. 27 (2021), no. 3, Paper No. 58, 8. MR 4273648, DOI 10.1007/s00041-021-09861-y
Aleksei Kulikov, Fedor Nazarov, and Mikhail Sodin, Fourier uniqueness and non-uniqueness pairs, arXiv:2306.14013 (2023).
David de Laat and Frank Vallentin, A breakthrough in sphere packing: the search for magic functions, Nieuw Arch. Wiskd. (5) 17 (2016), no. 3, 184–192. Includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Maryna Viazovska. MR 3643686
Rupert Li, Dual linear programming bounds for sphere packing via discrete reductions, arXiv:2206.09876 (2022).
Dalimil Mazáč and Miguel F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, J. High Energy Phys. 2 (2019), 163, front matter + 53. MR 3925259, DOI 10.1007/jhep02(2019)163
João P. G. Ramos and Mateus Sousa, Perturbed interpolation formulae and applications, arXiv:2005.10337 (2020).
João P. G. Ramos and Martin Stoller, Perturbed Fourier uniqueness and interpolation results in higher dimensions, J. Funct. Anal. 282 (2022), no. 12, Paper No. 109448, 34. MR 4403065, DOI 10.1016/j.jfa.2022.109448
Danylo Radchenko and Martin Stoller, Fourier non-uniqueness sets from totally real number fields, Comment. Math. Helv. 97 (2022), no. 3, 513–553. MR 4468993, DOI 10.4171/cmh/538
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
Naser Talebizadeh Sardari, Higher Fourier interpolation on the plane, arXiv:2102.08753 (2021).
Martin Stoller, Fourier interpolation from spheres, Trans. Amer. Math. Soc. 374 (2021), no. 11, 8045–8079. MR 4328691, DOI 10.1090/tran/8440
Thomas M. Thompson, From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, Mathematical Association of America, Washington, DC, 1983. MR 749038, DOI 10.5948/UPO9781614440215
A. Thue, Om nogle geometrisk-taltheoretiske Theoremer, Forhandlingerne ved de Skandinaviske Naturforskeres 14 (1892), 352–353.
Axel Thue, Über die dichteste Zusammenstellung von kongruenten Kreisen in der Ebene, Skrifter udgivne af Videnskabs-Selskabet i Christiania. I. Mathematisk-Naturvidenskabelig Klasse 1 (1910), 1–9.
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
Maryna Viazovska, Sharp sphere packings, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 455–466. MR 3966775, DOI 10.1142/9789813272880_0063
Maryna Viazovska, Almost impossible $E_8$ and Leech lattices, Eur. Math. Soc. Mag. 121 (2021), 4–8. MR 4400365, DOI 10.4171/mag-47