Common And Sidorenko Linear Equations
<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 ran...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Oxford University Press (OUP)
2022
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| Online Access: | https://hdl.handle.net/1721.1/145891 |
| _version_ | 1826216467028246528 |
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| author | Fox, Jacob Pham, Huy tuan Zhao, Yufei |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Fox, Jacob Pham, Huy tuan Zhao, Yufei |
| author_sort | Fox, Jacob |
| collection | MIT |
| description | <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> |
| first_indexed | 2024-09-23T16:48:10Z |
| format | Article |
| id | mit-1721.1/145891 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:48:10Z |
| publishDate | 2022 |
| publisher | Oxford University Press (OUP) |
| record_format | dspace |
| spelling | mit-1721.1/1458912022-10-19T03:01:34Z Common And Sidorenko Linear Equations Fox, Jacob Pham, Huy tuan Zhao, Yufei Massachusetts Institute of Technology. Department of Mathematics <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> 2022-10-18T16:57:18Z 2022-10-18T16:57:18Z 2021 2022-10-18T16:46:24Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/145891 Fox, Jacob, Pham, Huy tuan and Zhao, Yufei. 2021. "Common And Sidorenko Linear Equations." Quarterly Journal of Mathematics, 72 (4). en 10.1093/QMATH/HAAA068 Quarterly Journal of Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press (OUP) arXiv |
| spellingShingle | Fox, Jacob Pham, Huy tuan Zhao, Yufei Common And Sidorenko Linear Equations |
| title | Common And Sidorenko Linear Equations |
| title_full | Common And Sidorenko Linear Equations |
| title_fullStr | Common And Sidorenko Linear Equations |
| title_full_unstemmed | Common And Sidorenko Linear Equations |
| title_short | Common And Sidorenko Linear Equations |
| title_sort | common and sidorenko linear equations |
| url | https://hdl.handle.net/1721.1/145891 |
| work_keys_str_mv | AT foxjacob commonandsidorenkolinearequations AT phamhuytuan commonandsidorenkolinearequations AT zhaoyufei commonandsidorenkolinearequations |