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Small Sumsets in Free Products of \(\mathbb{Z}/2\mathbb{Z}\)

  • Shalom EliahouEmail author
  • Cédric Lecouvey
Chapter

Summary

Let G be a group. For positive integers r, s ≤ | G |, let μ G (r, s) denote the smallest possible size of a sumset (or product set) AB = { abaA, bB} for any subsets A, BG subject to | A | = r, | B | = s. The behavior of μ G (r, s) is unknown for the free product G of groups G i , except if the factors G i are all isomorphic to \(\mathbb{Z}\), in which case \({\mu }_{G}(r,s) = r + s - 1\) by a theorem of Kemperman for torsion-free groups (1956). In this paper, we settle the case of a free product G whose factors G i are all isomorphic to \(\mathbb{Z}/2\mathbb{Z}\), and prove that \({\mu }_{G}(r,s) = r + s - 2\) or \(r + s - 1\), depending on whether r and s are both even or not.

Keywords

Additive combinatorics Cauchy subsets Reduced words 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Univ Lille Nord de FranceLilleFrance
  2. 2.ULCO, LMPA Joseph LiouvilleCalaisFrance
  3. 3.FR CNRSValenciennesFrance

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