Diophantine Approximation with Piatetski-Shapiro Primes

Abstract

We prove that for every irrational number $\hat{I}\pm$, real number ${\hat{I}}^2$, real number $c$ satisfying $1<c<9/8$ and positive real number $\hat{I}¸$ satisfying $\hat{I}¸<(9/c-8)/10$, there exist infinitely many primes of the form $p=\left[n^c\right]$ with $n\in \mathbb{N}$ such that $||\hat{I}\pm p||<p^{-\hat{I}¸}$.

12 pages