Plünnecke's Inequality

Details Plünnecke's Inequality Plünnecke's Inequality

2011-01-13

arxiv.org/abs/1101.2532v3

Combinatorics, Probability and Computing, volume 20, issue 06, pp. 921-938, 2011

Mathematics

Combinatorics

19 pages 2 figures. Some structural changes and minor modifications

Scientific paper

Plunnecke's inequality is the standard tool to obtain estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be attained.

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