# Optimality of the Plünnecke-Ruzsa Inequality

The Plünnecke–Ruzsa Inequality states that for a finite subset $A$ of an abelian group $G$ with small doubling $|A+A|\le K|A|$, the iterated sum and difference sets are also small:

$|tA-sA| \le K^{t+s} |A|$.

It seems natural to expect that the optimal exponent on $K$ should actually be $K^{t+s-1}$, since we're thinking of each additional sum or difference with $A$ as magnifying the set by a factor of at most $K$.

Also, the choice of $A$ as a basis for $\mathbb{F}_2^n$ leads me to think there could be improvements on the constant in Plünnecke–Ruzsa, so that I would conjecture (roughly)

$|tA - sA| \le \frac{K^{t+s-1}}{t!s!} |A|$.

Is this possible?

I don't see a way to get improvements of either kind directly from the Plünnecke/Ruzsa/Petridis graph approach (see Petridis).

• From these I can find examples of $A$ where $|A-A|$ is much bigger than $|A+A|$, and also examples of $A,B$ where $|A+tB|$ is much bigger than $K^t |A|$. Are there examples where just the iterated sumset of a single set $|tA|$ is bigger than expected? – Xiaoyu He Oct 31 '17 at 2:47