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Bounds for sets with no polynomial progressions
Published online by Cambridge University Press: 05 January 2021
Abstract
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
- Type
- Number Theory
- Information
- Creative Commons
- 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
References
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