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Let $(G/\mathrm{\Gamma <\#comment/>},R_a)$be an ergodic $k$-step nilsystem for $k\ge <\#comment/>2$. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that $L^2\left(G/\mathrm{\Gamma <\#comment/>}\right)$decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math.124(2014), pp. 261–295] for $2$-step nilsystems and a result by Stepin [Uspehi Mat. Nauk24(1969), pp. 241–242] for nilsystems $G/\mathrm{\Gamma <\#comment/>}$with connected, simply connected $G$. 

Abstract We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if$$\Gamma $$is a countable discrete abelian group,$$\varphi , \psi \in \mathrm {End}(\Gamma )$$, and$$\psi - \varphi $$is an injective endomorphism with finite index image, then for any ergodic measure-preserving$$\Gamma $$-system$$( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$$, any measurable set$$A \in {\mathcal {X}}$$, and any$${\varepsilon }> 0$$, there is a syndetic set of$$g \in \Gamma$$such that$$\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$$. This generalizes the main results of Ackelsberget al[Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107]. For the group$$\Gamma = {\mathbb {Z}}^d$$, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to thequasi-affine(orConze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to$$\varphi $$and$$\psi $$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. 

Abstract The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $$\varphi , \psi : G \to G$$ are homomorphisms, such that at least two of the three subgroups $$\varphi (G)$$ , $$\psi (G)$$ and $$(\psi -\varphi )(G)$$ have finite index in G , then $$\{\varphi , \psi \}$$ has the large intersections property . That is, for any ergodic measure preserving system $$\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$$ , any $$A\in \mathcal {X}$$ and any $$\varepsilon>0$$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $$\varphi (g)=ag$$ and $$\psi (g)=bg$$ for $$a,b\in \mathbb {Z}$$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $$G=(\mathbb {Q}_{>0},\cdot )$$ and $$\varphi (g)=g$$ , $$\psi (g)=g^2$$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $$\varphi ,\psi $$ that have the large intersections property when $$G = {{\mathbb Z}}^2$$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor . In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $$\textbf {X}$$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $$\sigma $$ -algebras of $$\varphi (G)$$ and $$\psi (G)$$ invariant functions (Theorem 4.10).