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(n). 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 alpha(2) that provides a natural "threshold" for this distribution in the following sense. For any epsilon > 0, there exists a delta > 0 such that for any lattice, this distribution has mass at least d on [alpha(2) - epsilon, 1]; moreover, there exist lattices for which the distribution is tightly concentrated around alpha(2) (and so the mass on [alpha(2) + epsilon, 1] can be arbitrarily small). We also provide several bounds on alpha(2) and its extension to other l p norms. We end with an application from the area of computational complexity. Namely, we show that alpha(2) is exactly the approximation factor of a certain natural AM protocol for the Covering Radius Problem.
lattices covering radius computational complexity geometrical invariants second moment