Common and Sidorenko Linear Equations

IF 0.6 4区 数学 Q3 MATHEMATICS
Jacob Fox;Huy Tuan Pham;Yufei Zhao
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引用次数: 11

Abstract

A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $\mathbb{F}_q^{\,n}$ is asymptotically (as $n \to \infty$ ) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of $\mathbb{F}_q^{\,n}$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.
公线性方程和西多连科线性方程
系数为$\mathbb的线性方程{F}_q如果$\mathbb的任意两种着色中的单色溶液的数量为,则$是常见的{F}_q^{\,n}$至少是随机二着色中预期的数字。线性方程是Sidorenko,如果$\mathbb的任何稠密子集中的解的数量{F}_q^{\,n}$是渐近的,至少是在相同密度的随机集合中预期的数字。在本文中,我们刻画了那些常见的线性方程,以及那些是Sidorenko的线性方程。主要的新颖性是基于选择随机傅立叶系数的构造,该构造表明某些线性方程不具有这些性质。这解决了萨阿德和沃尔夫的一篇论文中提出的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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