Diophantine approximation by Piatetski-Shapiro primes

TL;DR

This paper proves the existence of infinitely many prime triples approximating a linear form with Piatetski-Shapiro primes, under certain conditions on parameters, extending Diophantine approximation results to a special class of primes.

Contribution

It establishes new Diophantine approximation results involving Piatetski-Shapiro primes for a range of c between 1 and 38/37, with explicit bounds and conditions.

Findings

Existence of infinitely many prime triples satisfying the approximation inequality.

Prime triples are of the form [n_i^c], with c in (1, 38/37).

Provides explicit bounds on the approximation error.

Abstract

Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $\eta$ is real, the constants $\lambda_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |\lambda_1p_1 + \lambda_2p_2 + \lambda_3p_3+\eta|<(\max p_j)^{{\frac{37c-38}{26c}}}(\log\max p_j)^{10} \end{equation*} and such that $p_i=[n_i^c]$, $i=1,\,2,\,3$.