# Bounds for sets with no nontrivial polynomial progressions

## Abstract/Contents

- Abstract
- In this thesis, we prove quantitative bounds in the polynomial Szemer\'edi theorem in several situations in which no bounds were previously known. In Chapter 2, we prove that if P_1, P_2\in\mathbb{Z}[y] are affine-linearly independent, then any subset of \mathbb{F}_q with no nontrivial polynomial progressions of the form x, x+P_1(y), x+P_2(y) must have size \ll_{P_1, P_2}q^{23/24}, provided the characteristic of \mathbb{F}_q is large enough. In Chapter 3, we prove that if P_1, \dots, P_m\in\mathbb{Z}[y] are affine-linearly independent, then any subset of \mathbb{F}_q with no nontrivial polynomial progressions of the form x, x+P_1(y), \dots, x+P_m(y) must have size \ll_{P_1, \dots, P_m}q^{1-\gamma_{P_1, \dots, P_m}} for some \gamma_{P_1, \dots, P_m}> 0, again provided that the characteristic of \mathbb{F}_q is large enough. In Chapter 4, we prove that any subset of \{1, \dots, N\} with no nontrivial progressions of the form x, x+y, x+y^2 must have size \ll N/(\log\log{N})^{2^{-157}}. In Chapter 5, we prove that if P_1, \dots, P_m\in\mathbb{Z}[y] have distinct degrees, then any subset of \{1, \dots, N\} with no nontrivial polynomial progressions of the form x, x+P_1(y), \dots, x+P_m(y) must have size \ll N/(\log\log{N})^{1-\gamma_{P_1, \dots, P_m}} for some \gamma_{P_1, \dots, P_m}> 0. In the final chapter, Chapter 6, we move to the nonabelian setting and prove power-saving bounds for subsets of nonabelian finite simple groups with no nontrivial progressions of the form x, xy, xy^2.

## Description

Type of resource | text |
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Form | electronic resource; remote; computer; online resource |

Extent | 1 online resource. |

Place | California |

Place | [Stanford, California] |

Publisher | [Stanford University] |

Copyright date | 2019; ©2019 |

Publication date | 2019; 2019 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Author | Peluse, Sarah Anne | |
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Degree supervisor | Soundararajan, Kannan, 1973- | |

Thesis advisor | Soundararajan, Kannan, 1973- | |

Thesis advisor | Fox, Jacob, 1984- | |

Thesis advisor | Manners, Frederick | |

Degree committee member | Fox, Jacob, 1984- | |

Degree committee member | Manners, Frederick | |

Associated with | Stanford University, Department of Mathematics. |

## Subjects

Genre | Theses |
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Genre | Text |

## Bibliographic information

Statement of responsibility | Sarah Peluse. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis Ph.D. Stanford University 2019. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2019 by Sarah Anne Peluse
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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