Optimally Small Sumsets in Groups, Ii. the Hypersmall Sumsets Property and Restricted Addition

This paper continues the work that was started in [Optimally small sumsets in groups, I. The supersmall sumsets property, the µ(k) and the G ν(k) functions, Uniform Distribution Theory 1 (2006), 27–43]. Here, we introduce G the hypersmall sumsets property and prove that it holds for all Abelian groups. As an application, we establish new upper bounds for the function ξG giving the minimal cardinality of a restricted sumset in an arbitrary Abelian group G. We then formulate the conjecture that these bounds are often optimal. Assuming a conjecture by Lev, we show that our conjecture is indeed very close to the truth. In some “generic” cases, we can even prove that it holds true.