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 $p_i=[n_i^c]$,  $i=1,\,2,\,3$.