In this context, A is a set, and A + A represents the set of all sums of pairs of elements drawn from A.
The paper doesn't use A^2, but rather |A|^2, which is ordinary squaring of an integer. It does use AA, which is the set of products of pairs of elements drawn from A.
The paper doesn't use A^2, but rather |A|^2, which is ordinary squaring of an integer. It does use AA, which is the set of products of pairs of elements drawn from A.