Introduction

There are many problems with a combinatorial flavour related to finite abelian groups. The first such result is most likely due to A. L. Cauchy [16] from 1813 and rediscovered by H. Davenport [25] in 1935. Let p be a prime number and let Z(p) be the additive group of residue classes modulo p. For two subsets A, B of Z(p) we define A + B to be {a + b : a ∈ A, b ∈ B} and we use |A| to denote the number of elements of A. The Cauchy-Davenport theorem now reads as follows. If A, B are not empty subsets of Z(p), then

|A+B| ≥ |A|+ |B| − 1, provided p ≥ |A|+ |B| − 1. This result has since been extended and generalized in a number of ways.