The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 In this paper we prove that no packing of unit balls in Euclidean space R8 has density greater than that of the E8 -lattice packing....
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The sphere packing problem in dimension 8 Maryna S. Viazovska April 5, 2017 arXiv:1603.04246v2 [math.NT] 4 Apr 2017 In this paper we prove that no packing of unit balls in Euclidean space R8 has density greater than that of the E8 -lattice packing. Keywords: Sphere packing, Modular forms, Fourier analysis AMS subject classification: 52C17, 11F03, 11F30 1 Introduction The sphere packing constant measures which portion of d-dimensional Euclidean space can be covered by non-overlapping unit balls. More precisely, let Rd be the Euclidean vector space equipped with distance k · k and Lebesgue measure Vol(·). For x ∈ Rd and r ∈ R>0 we denote by Bd (x, r) the open ball in Rd with center x and radius r. Let X ⊂ Rd be a discrete set of points such that kx − yk ≥ 2 for any distinct x, y ∈ X. Then the union [ P= Bd (x, 1) x∈X is a sphere packing. If X is a lattice in Rd then we say that P is a lattice sphere packing. The finite density of a packing P is defined as Vol(P ∩ Bd (0, r)) ∆P (r) := , r
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