Sphere packings in Euclidean space with forbidden distances

Gonçalves, Felipe  ; Vedana, Guilherme 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e49

Résumé

In this paper, we study the sphere packing problem in Euclidean space where we impose additional constraints on the separations of the center points. We prove that any sphere packing in dimension $48$, with spheres of radii r, such that no two centers $x_1$ and $x_2$ satisfy $\sqrt {\tfrac {4}{3}} < \frac {1}{2r}|x_1-x_2| <\sqrt {\tfrac {5}{3}}$, has center density less or equal than $(3/2)^{24}$. Equality occurs for periodic packings if and only if the packing is given by a $48$-dimensional even unimodular extremal lattice. This shows that any of the lattices $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ are optimal for this constrained packing problem, and gives evidence towards the conjecture that extremal lattices are optimal unconstrained sphere packings in $48$ dimensions. We also provide results for packings up to dimension $d\leq 1200$, where we impose constraints on the distance between centers and on the minimal norm of the spectrum, showing that even unimodular extremal lattices are again uniquely optimal. Moreover, in the one-dimensional case, where it is not at all clear that periodic packings are among those with largest density, we nevertheless give a condition on the set of constraints that allows this to happen, and we develop an algorithm to find these periodic configurations by relating the problem to a question about dominos.

Détail

DOI : 10.1017/fms.2025.9

Bibliographie

[1] Bassalygo, L., Cohen, G. and Zémor, , ‘Codes with forbidden distances’, Discrete Math. 213 (2000), 3–11. Google Scholar | DOI

[2] Bassalygo, L. A., Zinoviev, V. A., Zyablov, V. V., Pinsker, M. S. and Poltyrev, G. Sh., ‘Bounds for codes with unequal protection of two message sets’, Problemy Peredachi Informatsii 15(3) (1979), 40–49. Google Scholar

[3] Batut, C., Belabas, K., Benardi, D., Cohen, H. and Olivier, M., User’s Guide to PARI-GP, version 2.11.1 (2018). Google Scholar

[4] Berdnikov, A. V., ‘Estimate for the chromatic number of euclidean space with several forbidden distances’, Math. Notes 99(5) (2016), 774–778. Google Scholar | DOI

[5] Boyvalenkov, P. and Cherkashin, D., ‘The kissing number in 48 dimensions for codes with certain forbidden distances is 52 416 000’, Results in Mathematics 8(3) (2025). Google Scholar

[6] Boyvalenkov, P., Cherkashin, D. and Dragnev, P., ‘Universal optimality of T-avoiding spherical codes and designs’, Preprint, 2025, . Google Scholar | arXiv

[7] Boyvalenkov, P. and Dragnev, P., ‘Energy of codes with forbidden distances in 48 dimensions’, Preprint, 2024, . Google Scholar | arXiv

[8] Cohn, H. and Elkies, N., ‘New upper bounds on sphere packings I’, Ann. of Math. (2) 157(2) (2003), 689–714. Google Scholar | DOI

[9] Cohn, H. and Gonçalves, F., ‘An optimal uncertainty principle in twelve dimensions via modular forms’, Invent. Math. 217(3) (2019), 799–831. Google Scholar | DOI

[10] Cohn, H., Kumar, A., Miller, S., Radchenko, D. and Viazovska, M., ‘The sphere packing problem in dimension 24’, Ann. of Math. (2) 185(3) (2017), 1017–1033. Google Scholar | DOI

[11] Cohn, H., Kumar, A., Miller, S., Radchenko, D. and Viazovska, M., ‘Universal optimality of the and Leech lattices and interpolation formulas’, Ann. of Math. (to appear). Google Scholar

[12] Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups (Springer New York, 1999). Google Scholar | DOI

[13] Eichler, M. and Zagier, D., The Theory of Jacobi Forms (Progress in Mathematics) vol. 55 (Birkhäuser Boston, Inc., Boston, 1985). Google Scholar | DOI

[14] Enomoto, H., Frankl, P., Ito, N. and Nomura, K., ‘Codes with given distances’, Graphs Combin. 3 (1987), 25–38. Google Scholar | DOI

[15] Feigenbaum, A. S., Grabner, P. J. and Hardin, D. P., ‘Eigenfunctions of the Fourier transform with specified zeros’, Math. Proc. Cambridge Philos. Soc. 171(2) (2021), 329–367. Google Scholar | DOI

[16] Frankl, P., ‘Orthogonal vectors in the n-dimensional cube and codes with missing distances’, Combinatorica 6 (1986), 279–285. Google Scholar | DOI

[17] Groemer, H., ‘Existenzsätze für Lagerungen im Euklidischen Raum’, Math. Z. 81 (1963), 260–278. Google Scholar | DOI

[18] Hales, T. C., ‘A proof of the Kepler conjecture’, Ann. of Math. 162(3), (2005), 1065–1185. Google Scholar | DOI

[19] Jenkins, P. and Rouse, J., ‘Bounds for coefficients of cusp forms and extremal lattices’, Bull. London Math. Soc. 43(5) (2011), 927–938. Google Scholar | DOI

[20] Mallows, C. L., Odlyzko, A. M. and Sloane, N. J. A., ‘Upper bounds for modular forms, lattices, and codes’, J. Algebra 36 (1975), 68–76. Google Scholar | DOI

[21] Naslund, E., ‘The chromatic number of with multiple forbidden distances’, Mathematika 69(3) (2023), 692–718. Google Scholar | DOI

[22] Nebe, G., ‘A fourth extremal even unimodular lattice of dimension 48’, Discrete Math. 331 (2014), 133–136. Google Scholar | DOI

[23] Radchenko, D. and Viazovska, M., ‘Fourier interpolation on the real line’, Publ. Math. IHES 129 ((2019). 51–81. Google Scholar | DOI

[24] Raigorodskii, A. M., ‘The Borsuk problem and the chromatic numbers of some metric spaces’, Uspekhi Mat. Nauk 56 (2001), no. 1(337), 107–146. Google Scholar

[25] Rolen, L. and Wagner, I., ‘A note on Schwartz functions and modular forms’, Archiv der Mathematik 115 (2020), 35–51. Google Scholar | DOI

[26] Scharlau, R. and Schulze-Pillot, R., ‘Extremal lattices’, in Algorithmic Algebra and Number Theory (Springer, Berlin, Heidelberg, 1999), 139–170. Google Scholar | DOI

[27] Viazovska, M., ‘The sphere packing problem in dimension 8’, Ann. of Math. (2) 185(3) (2017), 991–1015. Google Scholar | DOI

[28] Zagier, D., ‘Elliptic modular forms and their applications’, in The 1-2-3 of Modular Forms (Universitext, Springer-Verlag, New York, 2008), 1–103. Google Scholar