论文标题
关于阿隆的问题
On a question of Alon
论文作者
论文摘要
$ \ mathbb {f} _p^n $中的线性方程式系统是\ textit {common}，如果每两色的$ \ mathbb {f} _p^n $的每两色至少与随机的单色解决方案相同，则与随机的两种颜色，非颜色，渐近，$ n \。通过类似于图理论设置，Alon询问是否通过添加足够多的自由变量来使线性方程的任何（非sidorenko）系统不常见。 Fox，Pham和Zhao在由单个方程式组成的系统之间的肯定中回答了这个问题。我们以负面的方式回答阿隆的问题。 我们还观察到，尽管添加了任意的许多自由变量，但仍然存在的特性与统一的概念密切相关，在这种概念中，人们用几何均值代替了单色溶液数量的算术平均值，并且可以解决Kamčev--liebenau-Morrison的问题。
A system of linear equations in $\mathbb{F}_p^n$ is \textit{common} if every two-colouring of $\mathbb{F}_p^n$ yields at least as many monochromatic solutions as a random two-colouring, asymptotically as $n \to \infty$. By analogy to the graph-theoretic setting, Alon has asked whether any (non-Sidorenko) system of linear equations can be made uncommon by adding sufficiently many free variables. Fox, Pham and Zhao answered this question in the affirmative among systems which consist of a single equation. We answer Alon's question in the negative. We also observe that the property of remaining common despite that addition of arbitrarily many free variables is closely related to a notion of commonness in which one replaces the arithmetic mean of the number of monochromatic solutions with the geometric mean, and furthermore resolve questions of Kamčev--Liebenau--Morrison.
