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 2 o ( 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 2 ο ( s ) in many cases. The main tool used in the proof is the asymmetric container lemma, introduced recently by Morris, Samotij and Saxton.

中文翻译：

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关于给定倍增常数的集合数

我们研究给定阿贝尔群 G 的 s 元素子集 J 的数量，使得 ∣ J + J ∣ ≤ K ∣ J ∣。证明 Alon、Balogh、Morris 和 Samotij 的猜想，并改进证明了 K 固定的猜想的 Green 和 Morris 的结果，我们提供了此类集合数量的上限，该上限紧至因子 2 o ( s ) ，当 G = ℤ 且 K = ο ( s /(log n ) 3 )。我们还将此结果推广到任意阿贝尔群，在许多情况下，该群紧至 2 ο ( s ) 的因数。证明中使用的主要工具是最近由 Morris、Samotij 和 Saxton 引入的非对称容器引理。