Abstract
Martin Kneser proved the following addition theorem for every abelian group G. If A, B ⊆ G are finite and nonempty, then \(\vert A + B\vert \geq \vert A + K\vert + \vert B + K\vert -\vert K\vert \) where \(K =\{ g \in G\mid g + A + B = A + B\}\). Here we give a short proof of this based on a simple intersection union argument.
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References
M. Kneser, Abschätzungen der asymptotischen Dichte von Summenmengen. Math. Z 58, 459–484 (1953)
M.B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, vol. 165 (Springer, New York, 1996)
T. Tao, V. Vu, Additive Combinatorics (Cambridge University Press, Cambridge, 2006)
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DeVos, M. (2014). A Short Proof of Kneser’s Addition Theorem for Abelian Groups. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, vol 101. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1601-6_3
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DOI: https://doi.org/10.1007/978-1-4939-1601-6_3
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