I was reading the proof of the Proposition 2.18 from Tao-Vu book but one moment is really confusing.
Question 1: I did not understand how the authors obtained ineqality which I highlighted by green color. But this is my guess: consider a function $f:M\to N$, where $$M=\{(x,x',c,c',a_1,a'_1)\in X\times X\times (A+B)\times (A+B)\times A\times A: z=c-a_1-x, \ z'=c'-a'_1-x'\}$$ and $$N=\{(x,x',c,c',a_1-a'_1)\in X\times X\times (A+B)\times (A+B)\times (A-A): z-z'=c-c'-(a_1-a'_1)-x+x'\}$$ defined by $(x,x',c,c',a_1,a'_1)\mapsto (x,x',c,c',a_1-a'_1)$.
It is obvious that $f$ is well-defined and injective. We already know that $|M|\geq \frac{|A|^2}{4}$. Since $f$ is injective, then $|N|\geq |M|$ and hence $|N|\geq \frac{|A|^2}{4}$. Is my reasoning correct?
If yes, then why they write the lines which I hightlighted by red color? what they want to say?
Question 2: Also I am slightly confused by the last paragraph of the proof. I understood that for any element of $2B-2B$ we can find $\geq \frac{|A|^2}{4}$ representations of the form $c-c'-d-x+x'$, where $(x,x',c,c',d)\in X\times X\times (A+B)\times (A+B)\times (A-A)$. How to obtain the desired inequality? Do I need to consider some function (injective or surjective?) $\phi: 2B-2B\to X\times X\times (A+B)\times (A+B)\times (A-A)$? Can anyone provide more details please?
