Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of last century, I.M. Vinogradov proved that inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes distance to a nearest integer. This result subsequently been improved by many authors. particular, Vaughan (1978) replaced exponent 1/5 1/4 using his celebrated identity for von Mangoldt function and refinement Fourier analytic arguments. The current record is due Matomäki (2009) who showed infinitude ||pα||≤p−1/3+ε. 1/3 considered limit technology. Recently, in [3], authors established an analogue Matomäki's on imaginary quadratic extensions field k=Fq(T). this paper, we consider case real k class number 1, which prove Vaughan's above-mentioned (exponent θ=1/4). Our method uses versions Dirichlet approximation theorem fields. latter was Arijit Ganguly appendix our previous paper [3] case. We also simplify arguments [1] same problem fields D. Mazumder first-named author.