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Improved estimates for polynomial Roth type theorems in finite fields

Dong Dong Xiaochun Li Will Sawin

Classical Analysis and ODEs mathscidoc:1910.43452

arXiv preprint arXiv:1709.00080, 2017.8
We prove that, under certain conditions on the function pair arphi_1 and arphi_1 , bilinear average arphi_1 along curve arphi_1 satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if arphi_1 with arphi_1 are linearly independent polynomials, then for any arphi_1 with arphi_1 , there are arphi_1 triplets arphi_1 . This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.
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@inproceedings{dong2017improved,
  title={Improved estimates for polynomial Roth type theorems in finite fields},
  author={Dong Dong, Xiaochun Li, and Will Sawin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020174758325284981},
  booktitle={arXiv preprint arXiv:1709.00080},
  year={2017},
}
Dong Dong, Xiaochun Li, and Will Sawin. Improved estimates for polynomial Roth type theorems in finite fields. 2017. In arXiv preprint arXiv:1709.00080. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020174758325284981.
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