和集有界的集合的典型近似结构
The typical approximate structure of sets with bounded sumset
摘要：
令$A_1$和$A_2$是基数$s_2\geq s_1 = Ω(s_2)$的第一个$n$整数的随机选择的子集，使得它们的和集$A_1+A_2$具有大小$m$。我们证明了渐近几乎必然$A_1$和$A_2$几乎完全包含在算术级数$P_1$和$P_2$中，具有相同的公差和基数近似$s_i m/(s_1+s_2)$。我们还证明了一个计数定理，这样的集合对在任意阿贝尔群。结果适用于$s_i = ω(\log^3 n)$和$s_1+s_2 \leq m = o(s_2/\log^3 n)$。我们的主要工具是一个不对称版本的超图容器的方法，最近使用的坎波斯证明类似的结果在特殊情况下$A=B$。
英文摘要：
Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers of cardinalities $s_2\geq s_1 = Ω(s_2)$, such that their sumset $A_1+A_2$ has size $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almost fully contained in arithmetic progressions $P_1$ and $P_2$ with the same common difference and cardinalities approximately $s_i m/(s_1+s_2)$. We also prove a counting theorem for such pairs of sets in arbitrary abelian groups. The results hold for $s_i = ω(\log^3 n)$ and $s_1+s_2 \leq m = o(s_2/\log^3 n)$. Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove similar results in the special case $A=B$.
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