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Plünnecke's Inequality


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.

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[1]Hall, P. (1935) On representatives of subsets. J. London Math. Soc. 10 2630.
[2]Malouf, J. L. (1995) On a theorem of Plünnecke concerning the sum of a basis and a set of positive density. J. Number Theory 54 1222.
[3]Menger, K. (1927) Zur allgemeinen Kurventheorie. Fund. Math. 10 96115.
[4]Nathanson, M. B. (1996) Additive Number Theory: Inverse Problems and the Geometry of Subsets, Springer.
[5]Plünnecke, H. (1970) Eine zahlentheoretische Anwendung der Graphtheorie. J. Reine Angew. Math. 243 171183.
[6]Ruzsa, I. Z. (1989) An application of graph theory to additive number theory. Scientia Ser. A 3 97109.
[7]Ruzsa, I. Z. (1990/1991) Addendum to: An application of graph theory to additive number theory. Scientia Ser. A 4 9394.
[8]Ruzsa, I. Z. (2009) Sumsets and structure. In Combinatorial Number Theory and Additive Group Theory, Springer.
[9]Tao, T. Additive combinatorics. Lecture notes 1, available online at
[10]Tao, T. and Vu, V. H. (2006) Additive Combinatorics, Cambridge University Press.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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