Optimally small sumsets in groups III. The generalized increasingly small sumsets property and the $\nu^{(k)}_G$} functions

Abstract

In this third part of our work, we go back to the study of the $\nu^{(k)}_G$ functions (introduced in the first one), which count the minimal cardinality of a sumset containing an element with a single representation. An upper bound for these functions is obtained in the case $k=2$ using what we call the generalized increasingly small sumsets property, which is proved to hold for all Abelian groups. Moreover, we show that our bound cannot be improved in general.