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arXiv · 1910.06436

Common and Sidorenko Linear Equations

Jacob Fox·Huy Tuan Pham·Yufei Zhao
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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.

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Jacob Fox, Huy Tuan Pham, Yufei Zhao. 2019-10-14. Common and Sidorenko Linear Equations. https://arxiv.org/abs/1910.06436

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