A note on the exceptional set for Diophantine approximation with mixed powers of primes

Abstract

Suppose that $\lambda _1, \lambda _2, \lambda _3, \lambda _4$ are nonzero real numbers, not all negative, and $\lambda _1/\lambda _2$ is irrational and algebraic. Let $\mathcal {V}$ be a well-spaced sequence, $\delta >0$. It is proved that for any $\varepsilon >0$, the number of $v\in {\mathcal {V}}$ with $v\le N$ for which

$$\begin{aligned} |\lambda _1p_1^2+\lambda _2p_2^3+\lambda _3p_3^4+\lambda _4p_4^5-v|<v^{-\delta } \end{aligned}$$

has no solution in primes $p_1, p_2, p_3, p_4$ does not exceed $O(N^{\frac{347}{360}+2\delta +\varepsilon })$. This result constitutes an improvement upon that of Ge and Zhao.