论文标题
在交替组的最大无产品子集上
On the Largest Product-free Subsets of the Alternating Groups
论文作者
论文摘要
如果对$ a = bc $没有$ a，b，c $的$ a = bc $的解决方案，则$ g $的子集$ a $称为无产品。很容易看出，对称组$ s_n $的最大无产品子集是通过获取所有奇数排列（即$ s_n \ setMinus a_n $）来获得的，其中$ a_n $是交替的组。相比之下，找到$ A_N $的最大无产品子集是一个长期的开放问题。我们解决了大型$ n $的问题，表明最大尺寸是通过先前猜想的极端示例来实现的，即形式的$ \ {π〜 | 〜π（x）\ in I，π（i）\ cap i = \ abternyset \} $及其inverses。此外，我们表明最大尺寸仅通过这些极端示例才能实现，并且我们具有稳定性：任何无产品的子集的$ a_n $的几乎最大尺寸的$ a_n $的子集在结构上接近极端示例。我们的证明使用了组合学和非亚伯傅里叶分析的工具组合，其中包括一种至关重要的新成分，利用了Filmus，Kindler，LiftShitz和Minzer开发的一些最新理论，用于对称组的全球超合同。
A subset $A$ of a group $G$ is called product-free if there is no solution to $a=bc$ with $a,b,c$ all in $A$. It is easy to see that the largest product-free subset of the symmetric group $S_n$ is obtained by taking the set of all odd permutations, i.e. $S_n \setminus A_n$, where $A_n$ is the alternating group. By contrast, it is a long-standing open problem to find the largest product-free subset of $A_n$. We solve this problem for large $n$, showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form $\{π~|~π(x)\in I, π(I)\cap I=\emptyset\}$ and their inverses. Moreover, we show that the maximum size is only achieved by these extremal examples, and we have stability: any product-free subset of $A_n$ of nearly maximum size is structurally close to an extremal example. Our proof uses a combination of tools from Combinatorics and Non-abelian Fourier Analysis, including a crucial new ingredient exploiting some recent theory developed by Filmus, Kindler, Liftshitz and Minzer for global hypercontractivity on the symmetric group.
