Basic Results for Sumsets with an Infinite Summand

Abstract

Recall that, when A and B are nonempty subsets of an abelian group G, we defined

$$|A+B|-|A|:=\sup\{|(A+B)\setminus(A+b)|:\, b\in B\}, $$

which, when A is finite, agrees with the usual definition, and when A is infinite, gives a natural definition of the previously undefined expression |A+B|−|A|=∞−∞. When A is finite, the supremum in the definition of |A+B|−|A| is irrelevant as |(A+B)∖(A+b)|=|(A+B)∖(A+b′)| for all b, b′∈B. The example A=[0,∞) and B=[0,n] in $\mathbb{Z}$ shows this is not true in general. This chapter is devoted to collecting together many of the basic results needed for dealing with sumsets having an infinite summand according to the above paradigm. The chapter culminates with the notion of restricted Freiman homomorphism, which is used to show how many questions with infinite summands can be directly reduced to the case when all summands are finite.