Hardy-Littlewood circle method applied to the binary problem

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I am having trouble understanding the application to this problem: binary I think this means the number of combinations of two numbers from the sets $A_1$ and $A_2$? Now Alessandro Zaccagnini goes on to show the solution (below) and I do not understand why $A$ x $A$ in $(a_1,a_2)\,\epsilon\, A$ x $A$? Why would the sets be multiplied?

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