Diophantine approximation with prime restriction in function fields

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