The exceptional set for Diophantine inequality with mixed powers of primes

Abstract

Assume that $\hat{I}{»}_1,\hat{I}{»}_2,\hat{I}{»}_3,\hat{I}{»}_4,\hat{I}{»}_5,\hat{I}{»}_6,\hat{I}{»}_7$ are non-zero real numbers , $\hat{I}{»}_1/\hat{I}{»}_2$ is an irrational number. Let $\mathcal{V}$ be a well-spaced sequence, and $\hat{I}´>0$. For any given positive integer $k\ge 5$ and any $\epsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V}$ with $\upsilon \le X$ for which the inequality $$\left|\hat{I}{»}_1{p}_1^2+\hat{I}{»}_2{p}_2^3+\hat{I}{»}_3{p}_3^3+\hat{I}{»}_4{p}_4^3+\hat{I}{»}_5{p}_5^3+\hat{I}{»}_6{p}_6^4+\hat{I}{»}_7{p}_7^k-\upsilon \right|<{\upsilon }^{-\hat{I}´}$$ has no solution in primes $p_1,p_2,p_3,p_4,p_5,p_6,p_7$.