Common And Sidorenko Linear Equations
Name
1910.06436.pdf
Description
Submitted version
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186.38 KB
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Checksum (MD5)
73ffc0b787ea497471b58debdc54c498
Author(s)
Fox, Jacob
Pham, Huy tuan
Zhao, Yufei
Date Issued
2021
Journal
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
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.
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1093/QMATH/HAAA068
