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Optimal Polynomial Recurrence


Published:20120412
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
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
\[\leftR(A,P,\varepsilon)\cap I\right
\geq c(\varepsilon,P)I\]
for all intervals $I$ of natural numbers with $I\geq L$ and
$c(\varepsilon,P)=\exp\exp(C\,\varepsilon^{1}\log\varepsilon^{1})$.