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Optimal Polynomial Recurrence
Published online by Cambridge University Press: 20 November 2018
Abstract
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Let 
$P\in \mathbb{Z}\left[ n \right]$ with 
$P(0)\,=\,0\,\text{and}\,\varepsilon \,>\,0$. We show, using Fourier analytic techniques, that if 
$N\ge \exp \exp \left( C{{\varepsilon }^{-1}}\log {{\varepsilon }^{-1}} \right)\,\text{and}\,A\,\subseteq \,\left\{ 1,\,.\,.\,.\,,\,N \right\}$ then there must exist 
$n\in \mathbb{N}$ such that
.
$$\frac{\left| A\cap \left( A+P\left( n \right) \right) \right|}{N}>{{\left( \frac{\left| A \right|}{N} \right)}^{2}}-\,\varepsilon $$
In addition to this we show, using the same Fourier analytic methods, that if 
$A\subseteq \mathbb{N}$, then the set
of 
$\varepsilon $-optimal return times
$$R\left( A,P,\varepsilon \right)=\left\{ n\in \mathbb{N}:\delta \left( A\cap \left. \left( A+P\left( n \right) \right) \right)> \right.\delta {{\left( A \right)}^{2}}-\varepsilon \right\}$$
is syndetic for every 
$\varepsilon >0$. Moreover, we show that 
$R\left( A,\,P,\,\varepsilon \right)$ is dense in every sufficiently long interval, in particular we show that there exists an 
$L=L\left( \varepsilon ,P,A \right)$ such that
$$\left| R\left( A,P,\varepsilon \right)\cap I \right|\ge c\left( \varepsilon ,P \right)\left| I \right|$$
for all intervals 
$I$ of natural numbers with 
$\left| I \right|\,\ge \,L\,\text{and}\,c\left( \varepsilon ,\,P \right)\,=\,\exp \exp \,\left( -C\,{{\varepsilon }^{-1}}\,\log {{\varepsilon }^{-1}} \right).$
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- Research Article
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- Copyright © Canadian Mathematical Society 2013
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