A Diophantine inequality with five squares of Piatetski-Shapiro primes

Abstract

Let $[\cdot ]$ denote the floor function. Assume that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$ are nonzero real numbers, not all of the same sign, that ${\lambda }_1/{\lambda }_2$ is irrational, and that $\eta$ is a real number. Let $\frac{71}{72}<\gamma <1$ and $\theta >0$. We prove that there exist infinitely many quintuples of primes $p_1,p_2,p_3,p_4,p_5$ satisfying the Diophantine inequality \begin{equation*} \big|λ_1p^2_1 + λ_2p^2_2 + λ_3p^2_3+ λ_4p^2_4 + λ_5p^2_5+η\big|<\big(\max p_j\big)^{\frac{71-72γ}{29}+θ}\,, \end{equation*} where $p_i=[n_i^{1/\gamma }]$, $i=1,2,3,4,5$. We also prove analogous theorems by raising the last variable in the inequality to the third and fourth powers.