Common And Sidorenko Linear Equations
Author(s)
Fox, Jacob; Pham, Huy tuan; Zhao, Yufei![Thumbnail](/bitstream/handle/1721.1/145891/1910.06436.pdf.jpg?sequence=4&isAllowed=y)
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<jats:title>Abstract</jats:title>
<jats:p>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.</jats:p>
Date issued
2021Department
Massachusetts Institute of Technology. Department of MathematicsJournal
Quarterly Journal of Mathematics
Publisher
Oxford University Press (OUP)
Citation
Fox, Jacob, Pham, Huy tuan and Zhao, Yufei. 2021. "Common And Sidorenko Linear Equations." Quarterly Journal of Mathematics, 72 (4).
Version: Original manuscript