Diophantine approximation by primes and Landau--Siegel zeros

Abstract

Let $\alpha>0$ be an irrational number of finite type and let $\beta\in\mathbb{R}$. Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\varepsilon>0$, there exist infinitely many primes $p$ such that \begin{gather*} \|\alpha p+\beta\|\leq p^{-1/3+\varepsilon} \qquad\text{(respectively, }\|\alpha p+\beta\|\leq p^{-1/3-1/20}\text{)}. \end{gather*}