Optimally small sumsets in groups III. the generalized increasingly small sumsets property and the v^(k) _G functions

In this third part of our work, we go back to the study of the v_G ^(k) functions (introduced in the flrst 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.