CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   LaTeX   MathJax   PDF  

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 Sarkozy, syndetic, polynomial return times
MSC Classifications: 11B30 show english descriptions Arithmetic combinatorics; higher degree uniformity 11B30 - Arithmetic combinatorics; higher degree uniformity
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/