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|\lambda_1p^2_1 + \lambda_2p^2_2 + \lambda_3p^2_3+ \lambda_4p^2_4 + \lambda_5p^2_5+\eta\big|<\big(\max p_j\big)^{\frac{71-72\gamma}{29}+\theta}\,, \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.