Sphere packings with forbidden distances

The Sphere Packing problem asks for the densest way of covering the euclidean space IR^d with congruent, solid balls. It was solved just for dimensions 1,2,3,8,24, the last two proved by Maryna Viazovska and her collaborators. In this talk we present a variation of this problem in which we forbid some short distances between centers of spheres. We prove that, under certain conditions, any sphere packing in 48 dimensions 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.