A Diophantine inequality involving different powers of primes of the form $[n^c]$

Abstract

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let $\frac{219}{220}<\gamma<1$ and $\theta>0$. We establish that, there exist infinitely many triples of primes $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |\lambda_1p_1 + \lambda_2p_2 + \lambda_3p^4_3+\eta|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220\gamma}{208}+\theta} \end{equation*} and such that $p_i=[n_i^{1/\gamma}]$, $i=1,\,2,\,3$.