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arXiv · 1909.00309

Bounds for sets with no polynomial progressions

Sarah Peluse
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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.

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Sarah Peluse. 2019-09-01. Bounds for sets with no polynomial progressions. https://doi.org/10.1017/fmp.2020.11

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