Diophantine approximation with a quaternary problem

Abstract

Let $1<k<7/6$, $\hat{I}{»}_1,\hat{I}{»}_2,\hat{I}{»}_3$ and $\hat{I}{»}_4$ be non-zero real numbers, not all of the same sign such that $\hat{I}{»}_1/\hat{I}{»}_2$ is irrational and let $\ddot{I}‰$ be a real number. We prove that the inequality $|\hat{I}{»}_1{p}_1^2+\hat{I}{»}_2{p}_2^2+\hat{I}{»}_3{p}_3^2+\hat{I}{»}_4{p}_4^k-\ddot{I}‰|\le ({\max }_j{p}_j{)}^{-\frac{7-6k}{14k}+\epsilon }$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\epsilon >0$.

arXiv admin note: text overlap with arXiv:1703.02381