What do Tao & Vu mean by "additive set"?

Question

On page 4 of Terence Tao & Van H. Vu's Additive Combinatorics there is the following theorem:

Let $A$ be an additive set of non-zero integers. Then $A$ contains a sum-free subset $B$ of size $|B| \gt |A|/3$.

How can you have an additive finite set that's interesting?

Answer 1

The prologue, on page xii, says:

An additive set is a pair $(A, Z)$, where $Z$ is an additive group, and $A$ is a finite non-empty subset of $Z$. We often abbreviate $(A,Z)$ as simply $A$, and refer to $Z$ as the ambient group of the additive set.

So every finite non-empty set of integers is an additive set.