IP⁎-sets in function field and mixing properties

The ring of polynomial over a finite field Fq[x] has received much attention, both from a combinatorial viewpoint as in regards to its action on measurable dynamical systems. In the case of (Z,+) we know that the ideal generated by any nonzero element is an IP⁎-set. In the present article we first establish that the analogous result is true for Fq[x]. We further use this result to establish some mixing properties of the action of (Fq[x],+). We shall also discuss on Khintchine's recurrence for the action of (Fq[x]∖{0},⋅).