Diophantine approximation with prime restriction in function fields

Abstract

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. We prove function field analogues of this result for the fields k=Fq(T) and imaginary quadratic extensions K of k. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.