A Diophantine inequality with five squares of Piatetski-Shapiro primes

Abstract: Let $[\,\cdot\,]$ denote the floor function. Assume that $λ_1, λ_2, λ_3, λ_4, λ_5$ are nonzero real numbers, not all of the same sign, that $λ_1/λ_2$ is irrational, and that $η$ is a real number. Let $\frac{71}{72}<γ<1$ and $θ>0$. 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 $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3,\,4,\,5$. We also prove analogous theorems by raising the last variable in the inequality to the third and fourth powers.