Diophantine Approximation with Piatetski-Shapiro Primes

Stephan Baier, Habibur Rahaman

Published 2024-08-02Version 1

We prove that for every irrational number $\alpha$, real number $c$ satisfying $1<c<9/8$ and positive real number $\theta$ satisfying $\theta<(9/c-8)/10$, there exist infinitely many primes of the form $p=\left[n^c\right]$ with $n\in \mathbb{N}$ such that $||\alpha p||<p^{-\theta}$.