Optimal Polynomial Recurrence

Lyall, Neil ; Magyar, Ákos

Canadian journal of mathematics, Tome 65 (2013) no. 1, pp. 171-194

Résumé

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 setof $\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 $\left| I \right|\,\ge \,L\,\text{and}\,c\left( \varepsilon ,\,P \right)\,=\,\exp \exp \,\left( -C\,{{\varepsilon }^{-1}}\,\log {{\varepsilon }^{-1}} \right).$

Détail

DOI : 10.4153/CJM-2012-003-8

Mots-clés : 11B30, Sarkozy, syndetic, polynomial return times