I was reading Ben Green's notes on additive combinatorics and there he writes the following proposition :
Suppose $U,V$ be subsets in some ambient abelian group $G$. Suppose that $U \sim V$. Then $U \sim -V$, $|U|\approx |V|$ and $\sigma[U]\approx 1$. We are dealing only with finite sets here. I have proved that $U \sim -V$ and $|U|\approx |V|$ but I am unable to prove that $\sigma[U]\approx 1$.
Definitions for reference: Let $x$ and $y$ be reals. Let $K\ge 2$ be some ambient parameter. We write $\displaystyle x <_{\approx} y$ if $x \leq K^cy$ for some $c$. We say $x\approx y$ iff $x <_{\approx} y$ and $y <_{\approx} x$.
Also, let $A$ and $B$ be two sets in some ambient abelian group and $K$ be some parameter. Then we write $A\sim B$ iff $\displaystyle \frac {|A-B|}{|A|^\frac{1}{2}|B|^\frac{1}{2}}\sim 1 $.
If U is a finite set, $\sigma[U]=|U+U|/|U|$.
Question: How can we prove that $\sigma[U]\approx 1$?
I don't really have any idea how to prove this. I just managed to note that $\sigma[U]=|U+U|/|U|\leq |U|^2/|U|=|U|$ and $\sigma[U]\ge 1$ but I think I need the bounds in powers of $K$. Also, I have not used the fact that $U\sim V$.
As Siming Tu has pointed out, the result does not seem to involve the original condition that $U\sim V$ and does not seem to be correct. Could anyone point to the right result?