IP*-sets in function field and mixing properties

The ring of polynomial over a finite field F-q[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 F-q[x]. We further use this result to establish some mixing properties of the action of (F-q[x],+) We shall also discuss on Khintchine's recurrence for the action of (F-q[x] \ {0},.). (c) 2017 Elsevier B.V. All rights reserved.