The main theorem of this paper says that no packing of unit balls in Euclidean space \(\mathbb{R}^8\) has density greater than \({\pi^4 / 384} \approx 0.25367\). This number is the density of the \(E_8\)-lattice packing \(\Lambda_8\). The proof is based on the technique of linear programming bounds. The crucial for the proof is a construction of two radial Schwartz functions \(a, b : \mathbb{R}^8 \to i \mathbb{R}\) which are eigenfunctions of the Fourier transform and have double zeroes at almost all points of \(\Lambda_8\). Moreover, the presented proof combined with some facts from the paper by H. Cohn and N. Elkies [Ann. Math. (2) 157, No. 2, 689–714 (2003; Zbl 1041.52011)] shows that the \(E_8\)-lattice packing is the unique periodic packing of maximal density.
Let us add that results of this paper by Viazovska are widely described and commented in the survey article by H. Cohn [Notices Am. Math. Soc. 64, No. 2, 102–115 (2017; Zbl 1368.52014)].