论文标题
线性方程的Sidorenko属性的本地方面
Local aspects of the Sidorenko property for linear equations
论文作者
论文摘要
$ \ Mathbb {f} _p^n $中的线性方程系统是\ textit {sidorenko}，如果$ \ mathbb {f} _p^n $的任何子集至少包含与系统的随机集合一样多的解决方案，则是相同密度的，差不多$ n \ n \ n \ forny \ forty \ forty。如果$ \ mathbb {f} _p^n $对线性方程式的系统是\ textIt {common}，那么对方程系统的单色求解与随机的2色，均为$ n \ to $ n \ to \ infty $。尽管最近关注，但两个分类问题仍然敞开。 我们表明，两个线性方程的某种通用系统不是Sidorenko。实际上，我们表明该家族中的系统不是本地的Sidorenko，并且该家族中不包含添加剂元组的系统并不是本地sidorenko的弱。这项工作回答了Kamčev-liebenau-Morrison的猜想和问题。就方法而言，我们观察到，线性系统的真正复杂性不能保持在傅立叶反转下。我们的主要新颖性是在具有复杂性的系统频率空间中使用高阶方法。我们还简短地证明了Kamčev-liebenau- morrison和独立的Versteegen的最新证明，任何包含四个任期算术进展的线性系统并不常见。
A system of linear equations in $\mathbb{F}_p^n$ is \textit{Sidorenko} if any subset of $\mathbb{F}_p^n$ contains at least as many solutions to the system as a random set of the same density, asymptotically as $n\to \infty$. A system of linear equations is \textit{common} if any 2-colouring of $\mathbb{F}_p^n$ yields at least as many monochromatic solutions to the system of equations as a random 2-colouring, asymptotically as $n\to \infty$. Both classification problems remain wide open despite recent attention. We show that a certain generic family of systems of two linear equations is not Sidorenko. In fact, we show that systems in this family are not locally Sidorenko, and that systems in this family which do not contain additive tuples are not weakly locally Sidorenko. This endeavour answers a conjecture and question of Kamčev--Liebenau--Morrison. Insofar as methods, we observe that the true complexity of a linear system is not maintained under Fourier inversion; our main novelty is the use of higher-order methods in the frequency space of systems which have complexity one. We also give a shorter proof of the recent result of Kamčev--Liebenau--Morrison and independently Versteegen that any linear system containing a four term arithmetic progression is uncommon.
