Optimally small sumsets in groups IV. Counting multiplicities and the λ_G functions

We continue our investigation on how small a sumset can be in a given abelian group. Here small takes into account not only the size of the sumset itself but also the number of elements which are repeated at least twice. A function λ_G(r, s) computing the minimal size (in this sense) of the sum of two sets with respective cardinalities r and s is introduced. (Lower and upper) bounds are obtained, which coincide in most cases. While upper bounds are obtained by constructions, lower bounds follow in particular from the use of a recent theorem by Grynkiewicz.