How to prove such inequality?

Question

I have some problems with proof of the inequality:

A - is a set.

I tried to use Plünnecke-Ruzsa inequality but didn't get any result. It will be great if someone help a little wuth this problem

Answer 1

This follows from induction on $k$ and the Ruzsa triangle inequality

$$ \lvert B\rvert \lvert A-C\rvert \leq \lvert A-B\rvert \lvert B-C\rvert. $$

The case $k=1$ is trivial. In general, we have

$$ \lvert A-2^k\cdot A\rvert \leq \frac{\lvert A-2^{k-1}\cdot A\rvert\lvert 2^{k-1}\cdot A-2^k\cdot A\rvert}{\lvert 2^{k-1}\cdot A\rvert},$$

and hence (since dilates preserve the cardinality of a set)

$$ \lvert A-2^k\cdot A\rvert \leq \lvert A-2^{k-1}\cdot A\rvert\frac{\lvert A-2\cdot A\rvert}{\lvert A\rvert}.$$

The result follows by induction.