Abstract
We study the number of s-element subsets J of a given abelian group G, such that ∣J + J∣ ≤ K∣J∣. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for K fixed, we provide an upper bound on the number of such sets which is tight up to a factor of 2o(s), when G = ℤ and K = ο(s/(log n)3). We also provide a generalization of this result to arbitrary abelian groups which is tight up to a factor of 2ο(s) in many cases. The main tool used in the proof is the asymmetric container lemma, introduced recently by Morris, Samotij and Saxton.
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Acknowledgements
The author would like to thank Rob Morris for his thorough comments on the manuscript and many helpful discussions. We would also like to thank Mauricio Collares, Victor Souza and Natasha Morrison for interesting discussions and comments on early versions of this manuscript. The author is also very grateful to Hoi Nguyen for spotting a mistake in and suggesting various improvements to the first version of this manuscript.
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Research partially supported by CNPq.
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Campos, M. On the number of sets with a given doubling constant. Isr. J. Math. 236, 711–726 (2020). https://doi.org/10.1007/s11856-020-1986-z
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DOI: https://doi.org/10.1007/s11856-020-1986-z