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*} |λ_1p_1 + λ_2p_2 + λ_3p^4_3+η|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220γ}{208}+θ} \end{equation*} and such that $p_i=[n_i^{1/\gamma }]$, $i=1,2,3$.