Piatetski-Shapiro primes in the intersection of multiple Beatty sequences

TL;DR

This paper proves the existence of infinitely many primes in the intersection of two Beatty sequences and a Piatetski-Shapiro sequence under certain conditions, extending prime distribution results in these specialized sequences.

Contribution

It establishes the first known result of infinitely many primes in the intersection of multiple Beatty sequences and a Piatetski-Shapiro sequence, with a sketch for multiple Beatty sequences.

Findings

Existence of infinitely many primes in the intersection under specified conditions

Conditions on irrationality and linear independence for sequences

Extension of prime distribution results to complex sequence intersections

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.