Diophantine approximation by Piatetski-Shapiro primes

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 $$\begin{aligned} |\lambda _1p_1 + \lambda _2p_2 + \lambda _3p_3+\eta |<(\max p_j)^{{\frac{37c-38}{26c}}}(\log \max p_j)^{10} \end{aligned}$$and such that \(p_i=[n_i^c]\), \(i=1,\,2,\,3\).