A note on Diophantine approximation with prime variables and mixed powers

Abstract

Let $k\ge 4$ be an integer. Suppose that $\lambda _1,\lambda _2,\lambda _3,\lambda _4$ are positive real numbers, $\frac{\lambda _1}{\lambda _2}$ is irrational and algebraic. Let $\mathcal {V}$ be a well-spaced sequence, and $\delta >0$. In this paper, we prove that, for any $\varepsilon >0$, the number of $\upsilon \in \mathcal {V}$ with $\upsilon \le X$ such that the inequality

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

has no solution in primes $p_1,p_2,p_3,p_4$ does not exceed $O(X^{1-\sigma ^*(k)+2\delta +\varepsilon })$, where $\sigma ^*(k)$ relies on k. This improves a recent result of Qu and Zeng (Ramanujan J 52:625–639, 2020).