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 ${\mathcal{N}}^{(c)}=\lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the intersection of multiple Beatty sequences.

24 pages