阿贝尔群中的Common方程和Sidorenko方程
Common and Sidorenko equations in Abelian groups
摘要：
如果对于$G$的每一个2-着色，该配置的单色实例的数目至少与随机选择的着色一样大，则称线性配置在有限阿贝尔群$G$中是公共的。Saad和Wolf指出，如果一个构形被定义为$G$上的一个齐次方程的解集，那么它在$\mathbb{F}_p^n$中是公共的，当且仅当方程的系数可以被划分为和为零mod $p$的对。Fox、Pham和Zhao在足够大的$n$中证明了这一点。我们将他们的结果推广到所有充分大的阿贝尔群$G$，其中方程的系数与$\vert G\vert$互质
英文摘要：
A linear configuration is said to be common in a finite Abelian group $G$ if for every 2-coloring of $G$ the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation over $G$, then it is common in $\mathbb{F}_p^n$ if and only if the equation's coefficients can be partitioned into pairs that sum to zero mod $p$. This was proven by Fox, Pham and Zhao for sufficiently large $n$. We generalize their result to all sufficiently large Abelian groups $G$ for which the equation's coefficients are coprime to $\vert G\vert$
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