Three-term polynomial progressions in subsets of finite fields
Three-term polynomial progressions in subsets of finite fields
复制标题
有限域子集中的三项多项式级数
DOI：
10.1007/s11856-018-1768-z
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发表时间：
2017
影响因子：
1
通讯作者：
Sarah Peluse
中科院分区：
文献类型：
--
作者：
Sarah Peluse
Bourgain and Chang recently showed that any subset of Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_p$$\end{document} of density ≫p−1/15 contains a nontrivial progression x, x + y, x + y2. We answer a question of theirs by proving that if P1, P2 ∈ ℤ[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_p$$\end{document} of density ≫P1,P2p−1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).
Bourgain and Chang recently showed that any subset of Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_p$$\end{document} of density ≫p−1/15 contains a nontrivial progression x, x + y, x + y2. We answer a question of theirs by proving that if P1, P2 ∈ ℤ[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_p$$\end{document} of density ≫P1,P2p−1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).