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Giorgis Petridis works on growth estimates in additive combinatorics. His paper gives a short proof of Plünnecke inequalities and extensions to product sets in groups.^[ref]

The proof chooses a subset with a minimal expansion ratio and uses that choice to control further sums. This converts one small-growth hypothesis into bounds for iterated sum-and-difference sets. The noncommutative setting requires separate product-set arguments because the order of multiplication matters.^[ref]

Let $A,B$ be nonempty finite subsets of an abelian group, and suppose $|A+B|\le K|A|$. For nonnegative integers $\ell ,m$,where $0B=\{0\}$. A Plünnecke inequality minimizing subset $X\subseteq A$ has $|X+jB|\le K^j|X|$ for every nonnegative $j$. The Ruzsa triangle inequality therefore givesThe empty-set exclusion is needed for the positive denominator and the growth hypothesis.^[ref]