On covering by translates of a set

TL;DR

This paper investigates the minimal number of translates needed to cover a group with a subset, analyzing efficiency, and generalizes classical results to broader contexts, especially for finite subsets in discrete groups.

Contribution

It extends classical covering results to broader settings and analyzes efficiency for finite subsets in discrete groups, including probabilistic aspects.

Findings

Efficiency is of order 1/log k for worst-case subsets

Almost all k-subsets cover the group efficiently when n is large

Classical results are generalized to broader group contexts

Abstract

In this paper we study the minimal number 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, we show that while the worst-case efficiency when $S$ has $k$ elements is of order $1/\log k$, for $k$ fixed and $n$ large, almost every $k$-subset of any given $n$-element group covers $G$ with close to optimal efficiency.