Optimal Polynomial Recurrence

http://dx.doi.org/10.4153/CJM-2012-003-8
Canad. J. Math. 65(2013), 171-194

  Published:2012-04-12
 Printed: Feb 2013

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 $c(\varepsilon,P)=\exp\exp(-C\,\varepsilon^{-1}\log\varepsilon^{-1})$.