Diophantine Approximation with Piatetski-Shapiro Primes

TL;DR

This paper proves that for certain irrational numbers and parameters, infinitely many primes of the form [n^c] approximate multiples of these irrationals with a specified precision, extending Diophantine approximation results to Piatetski-Shapiro primes.

Contribution

It establishes the existence of infinitely many Piatetski-Shapiro primes that approximate irrational multiples within a specific error bound, for a range of parameters.

Findings

Infinitely many primes of the form [n^c] approximate irrationals within p^{-θ}

Valid for c in (1, 9/8) and θ < (9/c - 8)/10

Extends Diophantine approximation to Piatetski-Shapiro primes.

Abstract

We prove that for every irrational number $\alpha$, real number $\beta$, 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 $||\alpha p||<p^{-\theta}$.