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Plünnecke's Inequality

Published online by Cambridge University Press:  05 October 2011

GIORGIS PETRIDIS*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: giorgis@cantab.net)

Abstract

Plünnecke's inequality is a standard tool for obtaining 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.

Type
Paper
Copyright
Copyright © Cambridge University Press 2011

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