Failure of Khintchine-type results along the polynomial image of IP${}_0$ sets

Abstract

In "IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,\mathcal{A},\mu ,T)$, any non-constant polynomial $p\in \mathbb{Z}[x]$ with $p(0)=0$, any $A\in \mathcal{A}$, and any $\epsilon >0$, the set $$R_{\epsilon }^p(A)=\{n\in \mathbb{N}|\mu (A\cap T^{-p(n)}A)>{\mu }^2(A)-\epsilon \}$$ is IP${}^{\ast }$, meaning that for any increasing sequence $(n_k{)}_{k\in \mathbb{N}}$ in $\mathbb{N}$, $$\text{FS}((n_k{)}_{k\in \mathbb{N}})\cap R_{\epsilon }^p(A)\ne \varnothing ,$$ where $$\text{FS}((n_k{)}_{k\in \mathbb{N}})=\{\sum _{j\in F}{n}_j|F\subseteq \mathbb{N}\text{ is finite}\text{ and }F\ne \varnothing \}=\{n_{k_1}+\cdots +n_{k_t}|k_1<\cdots <k_t,t\in \mathbb{N}\}.$$ In view of the potential new applications to combinatorics, this result has led to the question of whether a further strengthening of Khintchine's recurrence theorem holds, namely whether the set $R_{\epsilon }^p(A)$ is IP${}_0^{\ast }$ meaning that there exists a $t\in \mathbb{N}$ such that for any finite sequence $n_1<\cdots <n_t$ in $\mathbb{N}$, $$\{\sum _{j\in F}{n}_j|F\subseteq \{1,...,t\}\text{ and }F\ne \varnothing \}\cap R_{\epsilon }^p(A)\ne \varnothing .$$ In this paper we give a negative answer to this question by showing that for any given polynomial $p\in \mathbb{Z}[x]$ with deg$(p)>1$ and $p(0)=0$ there is an invertible probability preserving system $(X,\mathcal{A},\mu ,T)$, a set $A\in \mathcal{A}$, and an $\epsilon >0$ for which the set $R_{\epsilon }^p(A)$ is not IP${}_0^{\ast }$.

19 pages, referee comments implemented

References in corpus (1)

• The shifted primes and the multidimensional Szemeredi and polynomial van der Waerden theorems