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

TL;DR

This paper demonstrates that the strengthened form of Khintchine's recurrence theorem, involving IP$_0^*$ sets, does not hold for polynomial recurrence sets of degree greater than one, by providing explicit counterexamples.

Contribution

The paper shows that for polynomials of degree greater than one, the set of recurrence points can fail to be IP$_0^*$, disproving a potential strengthening of Khintchine's theorem.

Findings

Counterexamples for polynomials with degree > 1

Failure of IP$_0^*$ property in polynomial recurrence sets

Negative answer to the open question on strengthening Khintchine's recurrence

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$^*$, meaning that for any increasing sequence $(n_k)_{k\in\mathbb N}$ in $\mathbb N$, $$\text{FS}((n_k)_{k\in\mathbb N})\cap R_\epsilon^p(A)\neq \emptyset,$$ where $$\text{FS}((n_k)_{k\in\mathbb N})=\{\sum_{j\in F}n_j\,|\,F\subseteq \mathbb N\,\text{ is finite}\text{ and }F\neq\emptyset\}=\{n_{k_1}+\cdots+n_{k_t}\,|\,k_1<\cdots<k_t,\,t\in\mathbb N\}.$$ 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$_0^*$ meaning that there exists a $t\in\mathbb N$ such that for any finite sequence $n_1<\cdots<n_t$ in $\mathbb N$, $$\{\sum_{j\in F}n_j\,|\,F\subseteq \{1,...,t\}\text{ and }F\neq \emptyset\}\cap R_\epsilon^p(A)\neq \emptyset.$$ 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$_0^*$.