Abstract
The goal of this chapter is to prove the following basic result.
Theorem 5.1 (Pigeonhole and Multiplicity Bounds). Let G be an abelian group and let A, B⊆G be nonempty.
(i) If G is finite and |A|+|B|≥|G|+t, then r A,B (x)≥t for all x∈G.
(ii) If |A+B|<|A|+|B|−r+1, then r A,B (x)≥r for all x∈A+B.
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Grynkiewicz, D.J. (2013). The Pigeonhole and Multiplicity Bounds. In: Structural Additive Theory. Developments in Mathematics, vol 30. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00416-7_5
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DOI: https://doi.org/10.1007/978-3-319-00416-7_5
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