Exercise 2. 3. 8 from Tao-Vu's Additive Combinatorics

Question

Problem: Let $A$ be an additive set in an additive group $Z$ and let $G$ be a finite subgroup of $Z$. Show that $$\sigma[A + G] \leq \frac{|3A|}{|A|}.$$ Conclude that if $\pi: Z \to Z'$ is a group homomorphism then $$\sigma[\pi(A)] \leq \frac{|3A|}{|A|}.$$ The notation used is $\sigma[A] = \frac{|A+A|}{|A|}$ for an additive set $A$. The first part is easy but I cant find the solution to the second part of the problem. Any help is greatly appreciated.

Answer 1

The latter inequality can be proved directly, using the idea from the proof of the Ruzsa triangle inequality. In fact we can prove the more general fact

$$ \frac{\lvert \pi(B)\rvert}{\lvert \pi(A)\rvert}\leq \frac{\lvert A+B\rvert}{\lvert A\rvert},$$

(and then set $B=A+A$ to recover the desired inequality). To prove this, for any $z\in \pi(B)$, choose some $b_z$ arbitrarily such that $\pi(b_z)=z$. Then the map

$$ (z,a) \mapsto (a+b_z, \pi(a))$$

is easily checked to be an injection from $\pi(B)\times A \to (A+B)\times \pi(A)$.