Optimal Polynomial Recurrence

Abstract

Let $P\in \mathbb{Z}[n]$ with $P(0)=0$ and $\VE>0$. We show, using Fourier analytic techniques, that if $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq \{1,\dot{.}.,N\}$, then there must exist $n\in \mathbb{N}$ such that \[\frac{|A\cap (A+P(n))|}{N}>(\frac{|A|}{N})^2-\VE.\] In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq \mathbb{N}$, then the set of \emph{$\VE$-optimal return times} \[R(A,P,\VE)=\{n\in \N \,:\,\D(A\cap(A+P(n)))>\D(A)^2-\VE\}\] is syndetic for every $\VE>0$. Moreover, we show that $R(A,P,\VE)$ is \emph{dense} in every sufficiently long interval, in the sense that there exists an $L=L(\VE,P,A)$ such that \[|R(A,P,\VE)\cap I| \geq c(\VE,P)|I|\] for all intervals $I$ of natural numbers with $|I|\ge L$ and $c(\VE,P)=\exp\exp(-C\,\VE^{-1}\log\VE^{-1})$.

Short remark added and typos fixed

References in corpus (1)

• Simultaneous Polynomial Recurrence

Cited by in corpus (4)

• A Non-Linear Roth Theorem for Sets of Positive Density
• Polynomial progressions in topological fields
• Distances and Trees in Dense Subsets of ${\mathbb{Z}}^d$
• Bounds in a popular multidimensional nonlinear Roth theorem