The sphere packing problem in dimension \(24\)
The sphere packing problem is probably the most intriguing problem of discrete geometry. This problem is closely related to geometry, number theory and information theory (eg. optimal codes). Despite the numerous partial results in this field, up to 2016 the final solution was obtained only in dimensions 2 and 3. In [Ann. Math. (2) 185, No. 3, 991--1015 (2017; Zbl 1373.52025)], \textit{M. S. Viazovska} proposed a new breakthrough approach to sphere packing problem and proved that no packing of unit balls in Euclidean space \(\mathbb{R}^8\) has density greater than that of the \(E_8\)-lattice packing. Viazovska's idea was to use modular and quasimodular forms to construct radial eigenfunctions of Fourier transform in \(\mathbb{R}^8\) with eigenvalues \(\pm 1\) and to use some of their linear combination as auxiliary function in the Cohn-Elkies version of the linear programming bound. In this paper, the analogue of Viazovska's approach in the dimension 24 is presented. The main result states that the Leech lattice achieves the optimal sphere packing density in \(\mathbb{R}^{24}\), and it is the only periodic packing in \(\mathbb{R}^{24}\) with that density, up to scaling and isometries.
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