A Note on the Distribution of the Distance from a Lattice

Let L be an n-dimensional lattice and let x be a point chosen uniformly from a large ball in R. In this note we consider the distribution of the distance from x to L, normalized by the largest possible such distance (i.e., the covering radius of L). By definition, the support of this distribution is [0, 1]. We show that there exists a universal constant α2 that provides a natural ‘threshold’ for this distribution in the following sense. For any ε > 0 there exists a δ > 0 such that for any lattice this distribution has mass at least δ on [α2 − ε, 1]; moreover, there exist lattices for which the distribution is tightly concentrated around α2 (and so the mass on [α2 + ε, 1] can be arbitrarily small). We also provide several bounds on α2 and its extension to other `p norms. We end with an application from the area of computational complexity. Namely, we show that α2 is exactly the approximation factor of a certain natural AM protocol for the Covering Radius Problem.