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# Optimal Polynomial Recurrence

Published:2012-04-12
Printed: Feb 2013
• Neil Lyall,
Department of Mathematics, The University of Georgia, Athens, GA 30602, USA
• Ákos Magyar,
Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2
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## Abstract

Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$. We show, using Fourier analytic techniques, that if $N\geq \exp\exp(C\varepsilon^{-1}\log\varepsilon^{-1})$ and $A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that $\frac{|A\cap (A+P(n))|}{N}\gt \left(\frac{|A|}{N}\right)^2-\varepsilon.$ In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq\mathbb N$, then the set of $\varepsilon$-optimal return times $R(A,P,\varepsilon)=\left\{n\in \mathbb N \,:\,\delta(A\cap(A+P(n)))\gt \delta(A)^2-\varepsilon\right\}$ is syndetic for every $\varepsilon\gt 0$. Moreover, we show that $R(A,P,\varepsilon)$ is dense in every sufficiently long interval, in particular we show that there exists an $L=L(\varepsilon,P,A)$ such that $\left|R(A,P,\varepsilon)\cap I\right| \geq c(\varepsilon,P)|I|$ for all intervals $I$ of natural numbers with $|I|\geq L$ and $c(\varepsilon,P)=\exp\exp(-C\,\varepsilon^{-1}\log\varepsilon^{-1})$.
 Keywords: Sarkozy, syndetic, polynomial return times
 MSC Classifications: 11B30 - Arithmetic combinatorics; higher degree uniformity

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