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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Y.O. Hamidoune, Extensions of the Scherk–Kemperman Theorem. J. Comb. Theory, Ser. A 117, 974–980 (2010)
J.H.B. Kemperman, On complexes in a semigroup. Indag. Math. 18, 247–254 (1956)
J.H.B. Kemperman, On small sumsets in an abelian group. Acta Math. 103, 63–88 (1960)
V. Lev, Restricted set addition in Abelian groups: Results and conjectures. J. Théor. Nr. Bordx. 17, 181–193 (2005)
L. Moser, P. Scherk, Distinct elements in a set of sums. Am. Math. Mon. 62, 46–47 (1955)
H. Pan, Z.-W. Sun, Restricted sumsets and a conjecture of Lev. Isr. J. Math. 154(1), 21–28 (2006)
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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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