I am reading Tao-Vu book on Additive combinatorics and came across the following lemma. I know that it is better to ask this question on MathStack but I asked few questions before and no one answered yet and so I've decided to ask it here.
I understood most of the proof but I am slightly confused with the last part and how they obtained an upper bound in $(2.13)$? Can anyone explain it in a more detailed way please?
Thanks a lot!
EDIT: After some deep thoughts I came to the conclusion that authors do the following implicit construction: They define some function $\phi: (n+1)(A+B)\to A+B+nS$ which is surjective, then $$|A+B|^{n+1}\geq |(n+1)(A+B)|\geq|\phi^{-1}(A+B+nS)|=$$$$=\sum \limits_{x\in A+B+nS}|\phi^{-1}(\{x\})|\geq \sum \limits_{x\in A+B+nS} \left( \dfrac{|A||B|}{2|A+B|}\right)^n=|A+B+nS|\dfrac{|A|^n|B|^n}{2^n|A+B|^n}.$$
Hence we obtain $$|A+B+nS|\leq \dfrac{2^n|A+B|^{n+1}}{|A|^n|B|^n}.$$ But I do not know what is the function $\phi$.