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).