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

TL;DR

The paper addresses whether Khintchine-type recurrence for polynomial iterates can be strengthened to IP$_0^*$ along IP$_0$-images. It constructs Gaussian-system-based counterexamples via a Cantor-product spectral-measure framework that enforces negative correlations along finite-sum patterns, showing that for any polynomial $p$ with $p(0)=0$ and $ ext{deg}(p)>1$ the set $R_{oldsymbol{ u}}^p(A)$ need not be IP$_0^*$. The auxiliary spectral-measure construction (Sections 4.1–4.3) and the rigidity/weak-mixing realization (Section 5) underpin the main theorem, while the introduction discusses affirmative results under discrete spectrum or almost-IP contexts. Overall, the work delineates sharp limits on finitary recurrence phenomena along polynomial IP-images, even as IP$^*$-type results remain robust under broader conditions; it also demonstrates that Gaussian systems can exhibit both rigidity and weak mixing in this setting.

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 $(X,\mathcal A,μ,T)$, any non-constant polynomial $p\in\mathbb Z[x]$ with $p(0)=0$, any $A\in\mathcal A$, and any $ε>0$, the set $$R_ε^p(A)=\{n\in\mathbb N\,|\,μ(A\cap T^{-p(n)}A)>μ^2(A)-ε\}$$ is IP$^*$, meaning that for any increasing sequence $(n_k)_{k\in\mathbb N}$ in $\mathbb N$, $$\text{FS}((n_k)_{k\in\mathbb N})\cap R_ε^p(A)\neq \emptyset,$$ 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 $R_ε^p(A)$ is IP$_0^*$ meaning that there exists a $t\in\mathbb N$ such that for any finite sequence $n_1<\cdots<n_t$ in $\mathbb N$, $$\{\sum_{j\in F}n_j\,|\,F\subseteq \{1,...,t\}\text{ and }F\neq \emptyset\}\cap R_ε^p(A)\neq \emptyset.$$ 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 $(X,\mathcal A,μ,T)$, a set $A\in\mathcal A$, and an $ε>0$ for which the set $R_ε^p(A)$ is not IP$_0^*$.

Theorems & Definitions (27)

• Theorem 1.1: Cf. Corollary 2.1 in BFM
• Remark 1.2
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9: Cf. Theorem 1.8 in AlmostIPBerLeib
• Remark 2.1
• Lemma 2.2
• proof