Removing the finite-index hypotheses in the Khintchine-type multiple recurrence theorem for countable abelian groups

For a countable abelian group $G$ and integers $a,b$, Shalom's Khintchine-type recurrence theorem (proved independently by Ackelsberg, Bergelson and Best) states that if the four subgroups $aG$, $bG$, $(a-b)G$ and $(a+b)G$ all have finite index in $G$, then for every ergodic measure-preserving $G$-system, every measurable set $A$ and every $\varepsilon>0$, the set of $g\in G$ with $\mu(A\cap T_{ag}A\cap T_{bg}A\cap T_{(a+b)g}A)\ge\mu(A)^4-\varepsilon$ is syndetic. The author explicitly asks under which conditions on $a$ and $b$ these finite-index assumptions can be removed. The analogous question for 3-point configurations was subsequently settled by Ackelsberg, Bergelson and Shalom (one finite-index subgroup suffices for dilation pairs, with counterexamples when none exists), but the 4-point case is untouched. This problem asks for a characterization of the pairs $\{a,b\}$ for which the 4-point pattern $\{ag,bg,(a+b)g\}$ has the large intersections property in a countable abelian group: weaker sufficient conditions than the four finite-index hypotheses, and counterexamples delimiting where the property fails. The problem remains open.

Background

Khintchine's classical recurrence theorem asserts that for a single measure-preserving transformation $T$ of a probability space $(X,\mu)$ and any measurable set $A$ with $\mu(A)>0$, the set of return times $\{n\in\mathbb{Z}:\mu(A\cap T^{-n}A)>\mu(A)^2-\varepsilon\}$ is syndetic, meaning that it has bounded gaps (a subset $S$ of a group $G$ is syndetic if finitely many translates of $S$ cover $G$). Bergelson, Host and Kra proved the multiple-recurrence counterpart for ergodic systems: for the 3-term progression pattern the set $\{n:\mu(A\cap T^{-n}A\cap T^{-2n}A)>\mu(A)^3-\varepsilon\}$ is syndetic, and for the 4-point pattern $\{n,2n,3n\}$ the set $\{n:\mu(A\cap T^{-n}A\cap T^{-2n}A\cap T^{-3n}A)>\mu(A)^4-\varepsilon\}$ is syndetic, while for the patterns $\{n,2n,\dots,kn\}$ with $k\ge 4$ (five or more points) the corresponding set may be empty. In modern terminology, a family of homomorphisms has the large intersections property when such near-optimal lower bounds hold syndetically for every ergodic system.

The question extends naturally from $\mathbb{Z}$ to actions $(T_g)_{g\in G}$ of an arbitrary countable abelian group $G$, where the dilations $g\mapsto ag$ replace $n\mapsto kn$. Bergelson, Tao and Ziegler and later Shalom established the corresponding structure and recurrence theorems for vector spaces over finite fields and their direct sums. For general countable abelian groups, Shalom proved (independently of Ackelsberg, Bergelson and Best) that if the four subgroups $aG$, $bG$, $(a-b)G$ and $(a+b)G$ have finite index in $G$, then the pattern $\{ag,bg,(a+b)g\}$ has the large intersections property: for every ergodic $G$-system, every measurable $A$ and every $\varepsilon>0$, the set $\{g\in G:\mu(A\cap T_{ag}A\cap T_{bg}A\cap T_{(a+b)g}A)\ge\mu(A)^4-\varepsilon\}$ is syndetic. The proof identifies the characteristic factor of the associated multiple ergodic averages — the factor to which the limiting behaviour of the averages reduces — as a factor of the Conze–Lesigne factor, that is, the second level of the standard tower of Host–Kra–Ziegler characteristic factors, which under these hypotheses has the structure of a 2-step nilpotent homogeneous space (a quotient of a nilpotent Lie group of nilpotency class 2 by a lattice, equipped with Haar measure); the finite-index assumptions enter precisely to make this reduction available. They are automatic when $G=\mathbb{Z}$ (and more generally whenever $G$ is finitely generated) but restrictive for groups that are not finitely generated: for example, if $G=\bigoplus_{n\ge 1}\mathbb{Z}$ is a countable direct sum of copies of $\mathbb{Z}$, or $G=(\mathbb{Q}_{>0},\cdot)$ is the multiplicative group of positive rationals, which is isomorphic to such a direct sum with one copy of $\mathbb{Z}$ for each prime, then $aG$ has infinite index in $G$ for every integer $|a|\ge 2$. (By contrast, an additive divisible group such as $\mathbb{Q}$ satisfies $aG=G$ for every $a\neq 0$, so the hypotheses hold trivially there.)

For the 3-point pattern $\{ag,bg\}$ the parallel question has since been answered: Ackelsberg, Bergelson and Shalom proved that a single one of the subgroups $aG$, $bG$, $(b-a)G$ having finite index in $G$ suffices for the $\mu(A)^3$ bound, and, extending a counterexample of Chu, that the large intersections property fails in general when all three subgroups have infinite index. This isolates the 4-point pattern $\{ag,bg,(a+b)g\}$ as the unresolved case at the endpoint of the positive range.

Problem Statement

Let $G$ be a countable abelian group and let $a,b\in\mathbb{Z}$ be distinct nonzero integers with $a+b\neq 0$; these exclusions remove only degenerate patterns, since if $a\in\{0,b\}$ or $a+b=0$ the intersection below involves at most three distinct translates of $A$, and the resulting 2- and 3-point questions are already settled — the 2-point case by the classical Khintchine theorem and the 3-point case by the characterization of dilation pairs of Ackelsberg, Bergelson and Shalom recalled in the background. Say that the 4-point pattern $\{ag,bg,(a+b)g\}$ has the large intersections property in $G$ if for every ergodic measure-preserving $G$-system $(X,\mathcal{B},\mu,(T_g)_{g\in G})$, every measurable $A\in\mathcal{B}$ with $\mu(A)>0$, and every $\varepsilon>0$, the set

$$\{g\in G:\ \mu(A\cap T_{ag}A\cap T_{bg}A\cap T_{(a+b)g}A)\ge\mu(A)^4-\varepsilon\}$$

is syndetic in $G$. Shalom's theorem gives this under the hypothesis that all four subgroups $aG$, $bG$, $(a-b)G$ and $(a+b)G$ have finite index in $G$; the question, posed explicitly in the source, is under which conditions on $a$ and $b$ (relative to $G$) these hypotheses can be removed. The problem is to determine, for every countable abelian group $G$, exactly which pairs $\{a,b\}$ have the large intersections property: to characterize the dilation pairs for which the pattern $\{ag,bg,(a+b)g\}$ enjoys the syndeticity conclusion above, by establishing sufficient conditions on $a$ and $b$ and by exhibiting the ergodic systems and sets that show where the property fails. By the ergodic correspondence principle of Bergelson and Ferré Moragues, every sufficient condition established this way also yields the corresponding combinatorial statement, in the manner of the density theorem derived in the source: for every set $E\subseteq G$ and every $\varepsilon>0$, the set $\{g\in G:\ d_\Phi(E\cap E-ag\cap E-bg\cap E-(a+b)g)\ge d_\Phi(E)^4-\varepsilon\}$ is syndetic, where $d_\Phi$ is the upper density along a Følner sequence (a sequence of finite sets asymptotically invariant under translation).

References

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