Diophantine approximation with mixed powers of Piatetski-Shapiro primes

Abstract

Let $[\cdot ]$ denote the floor function. In this paper, we show that whenever $\eta$ is real and the constants ${\lambda }_i$ satisfy some necessary conditions, then for any fixed $\frac{63}{64}<\gamma <1$ and $\theta >0$, there exist infinitely many prime triples $p_1,p_2,p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^2_3+η|<\big(\max \{p_1, p_2, p^2_3\}\big)^{{\frac{63-64γ}{52}}+θ} \end{equation*} and such that $p_i=[n_i^{1/\gamma }]$, $i=1,2,3$.