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Bounds for sets with no polynomial progressions

Part of: Sequences and sets

Published online by Cambridge University Press:  05 January 2021

Sarah Peluse
Sarah Peluse*
Affiliation:
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ08540, USA; E-mail: speluse@princeton.edu

Abstract

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Let $P_1,\dots ,P_m\in \mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log {N})^{c_{P_1,\dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Type
Number Theory
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e16
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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