On recurrence in positive characteristic

Let P−1P−1 denote the set of primes minus 11. A classical theorem of A Sárkőzy says that any set of natural numbers of positive density contains a pair of elements whose difference belongs to P−1P−1. An ergodic approach to questions of this type was given by the fourth author, building on work of H. Furstenberg. In this paper we give a proof of the positive characteristic analogue of this result using the same approach.