Sumset magnification ratio strictly smaller for subset.

Question

Do there exist sets $X \subset A \subset \mathbb{Z}$ such that $$\frac{|A+X|}{|X|} < \frac{|A+A|}{|A|} $$?

I would also be happy if one can replace $\mathbb{Z}$ with any other abelian group.

Answer 1

This is a natural question to ask when reading Petridis's new argument of Plunnecke's inequality. The question has been asked and answered (with a counterexample) on mathoverflow and can be found here:

https://mathoverflow.net/questions/160207/a-sumset-inequality/160327#160327