On prime-producing sieves and distribution of $αp-β$ mod $1$

TL;DR

The paper proves that for any irrational $\alpha$ and real $\beta$, there are infinitely many primes $p$ satisfying $||\alpha p - \beta|| < p^{-\frac{28}{87}}$, improving previous bounds and linking to prime-producing sieves. The author adopts a Ford–Maynard type sieve framework with Type-I/II inputs, leveraging Buchstab's identity and a sequence of role-reversals to optimize the exponent, yielding $τ=\frac{28}{87}$ and $δ=(2x)^{-τ}$. A key contribution is identifying the triple $(\gamma,\theta,\nu)=(\frac{59}{87},\frac{28}{87},\frac{1}{29})$ that can produce primes with this property, while achieving minimal Type-II information with $\nu=\frac{1}{29}$. The work also yields corollaries on the distribution of $p^{\theta}-\beta$ mod 1 and on Diophantine approximation with Gaussian primes, illustrating the sieve's reach beyond the original problem. Overall, it advances the application of prime-producing sieves to sharp bounds in additive Diophantine approximation and related prime-distribution questions.

Abstract

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

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof