On Covering by Translates of a Set
In this paper we study the minimal number tau(S, G) of translates of an arbitrary subset S of a group G needed to cover the group, and related notions of the efficiency of such coverings. We focus mainly on finite subsets in discrete groups, reviewing the classical results in this area, and generalizing them to a much broader context. For example, the worst-case efficiency when S has k elements is of order 1/log k. We show that if n(k) grows at a suitable rate with k, then almost every k-subset of any given group with order n comes close to this worst-case bound. In contrast, if n(k) grows very rapidly, or if k is fixed and n ->infinity, then almost every k-subset of the cyclic group with order n comes close to the optimal efficiency. (C) 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 33-67, 2011
coverings tilings