Abstract
LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev.
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This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.
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Pan, H., Sun, ZW. Restricted sumsets and a conjecture of Lev. Isr. J. Math. 154, 21–28 (2006). https://doi.org/10.1007/BF02773597
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DOI: https://doi.org/10.1007/BF02773597