Multiple and Polynomial Recurrence for Abelian Actions in Infinite Measure

We apply the (C, F )-construction from [Da] to produce a number of funny rank one infinite measure preserving actions of Abelian groups G with “unusual” multiple recurrence properties. In particular, we construct the following for each p ∈ N ∪ {∞}: (i) a p-recurrent action T = (Tg)g∈G such that (if p 6=∞) no one transformation Tg is (p + 1)-recurrent for every element g of infinite order, (ii) an action T = (Tg)g∈G such that for every finite sequence g1, . . . , gr ∈ G without torsions the transformation Tg1 × · · · × Tgr is ergodic, p-recurrent but (if p 6=∞) not (p + 1)-recurrent, (iii) a p-polynomially recurrent (C, F )-transformation which (if p 6= ∞) is not (p + 1)-recurrent. ∞recurrence here means multiple recurrence. Moreover, we show that there exists a (C, F )-transformation which is rigid (and hence multiply recurrent) but not polynomially recurrent. Nevertheless, the subset of polynomially recurrent transformations is generic in the group of infinite measure preserving transformations endowed with the weak topology.