The author proves that there are infinitely many primes $p$ such that $\| \alpha p - \beta \| < p^{ -\frac{ 28}{ 87}}$, where $\alpha$ is an irrational number and $\beta$ is a real number. This sharpens a result of Jia (2000) and provides a new triple $(\gamma, \theta, \nu)=(\frac{ 59}{ 87}, \frac{ 28}{ 87}, \frac{ 1}{ 29})$ that can produce special primes in Ford and Maynard's work on prime-producing sieves. Our minimum amount of Type-II information required ($\nu = \frac{ 1}{ 29}$) is less than any previous work on this topic using only traditional Type-I and Type-II information.
Categories: math.NT