The distribution of $\alpha p$ modulo one in real quadratic function fields

Matomäki proved that if $\alpha \in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $\mathrm{\mid }\alpha -a\mathrm{/}p\mathrm{\mid }\le p^{-4\mathrm{/}3+\epsilon }$ for a suitable integer $a$. In \cite{BaMo}, Baier and the author established analogues of this result for ${\text{F}}_q(T)$ and its imaginary quadratic extensions. In this paper, we extend it to all real quadratic extensions of ${\text{F}}_q(T)$.