MULTIPLE ERGODIC AVERAGES IN ABELIAN GROUPS AND KHINTCHINE TYPE RECURRENCE

Let G be a countable abelian group. We study ergodic averages associated with configurations of the form {ag, bg, (a + b)g} for some a, b ∈ Z. Under some assumptions on G, we prove that the universal characteristic factor for these averages is a factor (Definition 1.15) of a 2-step nilpotent homogeneous space (Theorem 1.18). As an application we derive a Khintchine type recurrence result (Theorem 1.3). In particular, we prove that for every countable abelian group G, if a, b ∈ Z are such that aG, bG, (b − a)G and (a + b)G are of finite index in G, then for every E ⊂ G and ε > 0 the set {g ∈ G : d(E ∩ E − ag ∩ E − bg ∩ E − (a + b)g) ≥ d(E)⁴ − ε} is syndetic. This generalizes previous results for G = Z, G = F^ω_p and G = ⊕ _p∈P F_p by Bergelson, Host and Kra [Invent. Math. 160 (2005), pp. 261–303], Bergelson, Tao and Ziegler [J. Anal. Math. 127 (2015), pp. 329–378], and the author [Host-Kra theory for ⊕ _p∈P F_p-systems and multiple recurrence, arXiv:2101.04613.], respectively.