Common and Sidorenko Linear Equations. (20th January 2021)

• Record Type: Journal Article
• Authors: Fox, Jacob; Pham, Huy Tuan; Zhao, Yufei
• Is Part Of: Quarterly journal of mathematics. Volume 72:Part 4(2021)
• Page Start: 1223
• Page End: 1234
• Publication Date: 2021-01-20
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/qmath/haaa068
• Languages: English
• ISSNs: 0033-5606

Abstract

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.