 Solution

Apply Plünnecke's inequality with the summand set also equal to $A$. There is a nonempty $X\subseteq A$ satisfying $|X+mA|\le C^m|X|$ for every $m\ge 0$.

We need the Ruzsa triangle inequality in a form whose short proof also fixes the signs. For every $z\in R-S$, choose one representation $z=r_z-s_z$. The mapis injective into $(R+X)\times (S+X)$: its difference recovers $z$, then the fixed representation recovers $x$. ThereforeUse $R=rA$ and $S=sA$ and the common set $X$ from the previous part:ThusThe shared minimizing subset is important; applying separate growth estimates with unrelated subsets would not justify this bound.