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](https://link.springer.com/content/pdf/10.1007/978-1-4939-1601-6_16.pdf)).



Optimality of the Plünnecke-Ruzsa Inequality
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](https://zbmath.org/?q=an:1179.11002).


See also a number of papers of Ruzsa constructing various counterexamples, e.g.


*Ruzsa, I.Z.*, [**On the number of sums and differences**](http://dx.doi.org/10.1007/BF00050906), Acta Math. Hung. 59, No.3-4, 439-447 (1992). [ZBL0773.11010](https://zbmath.org/?q=an:0773.11010).


