TITLE: Optimality of the Plünnecke-Ruzsa Inequality
QUESTION [14 upvotes]: 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).

REPLY [18 votes]: No.  See for instance Exercise 2.3.5 of 
Tao, Terence; Vu, Van H., Additive combinatorics, Cambridge Studies in Advanced Mathematics 105. Cambridge: Cambridge University Press (ISBN 978-0-521-13656-3/pbk). xviii, 512 p. (2010). ZBL1179.11002.
See also a number of papers of Ruzsa constructing various counterexamples, e.g.
Ruzsa, I.Z., On the number of sums and differences, Acta Math. Hung. 59, No.3-4, 439-447 (1992). ZBL0773.11010.