A Note on the Distribution of the Distance from a Lattice

Abstract

Let ℒ be an n-dimensional lattice, and let x be a point chosen uniformly from a large ball in ℝn. 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 α 2 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 [α 2−ε,1]; moreover, there exist lattices for which the distribution is tightly concentrated around α 2 (and so the mass on [α 2+ε,1] can be arbitrarily small). We also provide several bounds on α 2 and its extension to other ℓ p norms. We end with an application from the area of computational complexity. Namely, we show that α 2 is exactly the approximation factor of a certain natural \(\mathsf{AM}\) protocol for the Covering Radius Problem.