Piatetski-Shapiro primes in the intersection of multiple Beatty sequences

Abstract

Suppose that $\alpha_1, \alpha_2,\beta_1, \beta_2 \in\mathbb{R}$. Let $\alpha_1, \alpha_2 > 1$ be irrational and of finite type such that $1, \alpha_1^{-1}, \alpha_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number in the range $1 < c < 12/11$. In this paper, it is proved that there exist infinitely many primes in the intersection of Beatty sequences $\mathcal{B}_{\alpha_1,\beta_1} = \lfloor\alpha_1 n + \beta_1\rfloor, \mathcal{B}_{\alpha_2, \beta_2} = \lfloor\alpha_2 n + \beta_2\rfloor$ and the Piatetski-Shapiro sequence $\mathscr{N}^{(c)} = \lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the intersection of multiple Beatty sequences.